Model Theory of Absolute Galois Groups

Home , Absolute Galois group

Dissertation

zur Erlangung des Doktorgrades der Fakult¨at fur¨ Mathematik und Physik der Albert–Ludwigs–Universit¨at Freiburg

vorgelegt von

Nina Frohn

Freiburg, Januar 2011 Dekan: Prof. Dr. Kay K¨onigsmann

Erster Gutachter: Dr. Jochen Koenigsmann Zweiter Gutachter: Prof. Dr. Martin Ziegler

Datum der Promotion: 10.12.2010

Abteilung fur¨ Mathematische Logik Fakult¨at fur¨ Mathematik und Physik Albert-Ludwigs-Universit¨at Freiburg Eckerstraße 1 79104 Freiburg i. Br. Introduction

Galois theory is a lively area of mathematical research in which several branches of mathematics (as algebra, number theory and geometry) are involved. This thesis is concerned with the model theoretic aspects of Galois theory, and in particular with the model theoretic aspects of the universal object of Galois theory, the absolute Galois group. Model theory often studies elementary classes of first-order structures, or a class of first-order structures sharing some model-theoretic property. The central question in the background of this work is the question whether or not the class of absolute Galois groups is an elementary class in an appropriate first- order language. The answer is likely to be negative: The general group theoretic structure of absolute Galois groups is not known very well, and there is no group theoretical characterization of those profinite groups which occur as absolute Galois groups, not even conjecturally. The results in this thesis can hopefully be used to give a negative answer to this question in the future. Having this question as a central motivation in mind, this thesis deals with the following problems: • Which first order language should be chosen if one wants to set up a model theory of absolute Galois groups, or more general of profinite groups? The answer to this question is not as obvious as one might think: The usual group language may come to one’s mind first, but it has the disadvantage of not capturing the topological structure of the group. An approach that is better adapted to profinite groups is describing the inverse system of the group instead of the group itself. • Which subclasses of the class of all profinite groups are elementary? For example, in Chapter 5 we give a complete classification of the elementary theories of abelian profinite groups. • Which properties shared by all absolute Galois groups are axiomatizable, and what is the elementary theory of the class of absolute Galois groups? We will investigate various properties of absolute Galois groups and check them for axiomatizability. • Are there natural subclasses of the class of all absolute Galois groups from which we can prove or disprove that they are elementary? For example we will show that for every fixed natural number n, the class of maximal pro-p quotients of absolute Galois groups of rank n is elementary, see Corollary 3.7. In the eighties, G. Cherlin, L. van den Dries and A. Macintyre introduced a first- order language in which they considered profinite groups as model theoretic

i structures, see [CDM]. Their attempt was based on substituting the original profinite group by its associated inverse system, and describing this in an appropriate first-order language. The preprint [CDM] contains already first results about absolute Galois groups, for example that the absolute Galois group of K is interpretable in K, and that the class of absolute Galois groups is closed under ultraproducts. There is later and more extensive work on the model theory of profinite groups by Z. Chatzidakis, see for example [C1] and [C2]. But to this day, the model theory of profinite groups is not very elaborated, and so big parts of this thesis are concerned with the development of the model theory of arbitrary profinite groups. An example: To work towards a proof for the conjecture that the class of absolute Galois groups is non-elementary, we need to understand the conditions under which absolute Galois groups are elementary substructures of each other. For this purpose, we first need to understand elementary equivalence in arbitrary profinite groups. Other parts of this thesis are concerned with the algebraic structure of absolute Galois groups not including any model theoretic content. The reason is the following: Studying the model theory of absolute Galois groups often calls one’s attention to questions concerning the algebraic analysis of absolute Ga- lois groups. Sometimes solving this algebraic question is just a necessary duty before proceeding to the model theoretic problem behind it, but sometimes the algebraic question is worth a discussion on its own sake. An example is Chap- ter 4 about Demushkin groups: The study of these groups in connection with the questions about axiomatizability in Section 3.4 eventually led to the purely algebraic results stated in Chapter 4. Finally, the two aspects “necessary preliminary work” and “interesting side product” can melt together as in Chapter 5: The characterization of abelian profinite groups given there is intended to be used to construct a pair G1 ≺ G2 of profinite groups, such that G2 is an absolute Galois group, and such that the maximal abelian quotient of G1 guarantees that G1 cannot be an absolute Galois group, compare the considerations on page 62. But the classification is very interesting in itself and leads to a satisfying result concerning the model theory of abelian profinite groups. This thesis is organized as follows: In Chapter 1, the necessary foundations for the thesis are provided: Basics about profinite groups, with a special emphasis on the cohomology of profinite groups and on Brauer groups; some fundamental facts about (abstract) abelian groups; and finally some standard valuation theory. Chapter 2 is concerned with the model theory of arbitrary profinite groups. We will mostly work in the language LIS introduced in [CDM] describing the inverse system of the profinite group G, but we will also compare the expressive power of LIS with the one of the usual group language LG. Moreover we develop a characterization of LIS-elementary equivalence of profinite groups by means of a modified Ehrenfeucht-Fraisségame which will be used in Chapter 5. In Chapter 3, we will discuss the axiomatizability of some properties P shared by all absolute Galois groups. The difficulty of this procedure is that the property P might not be defined by group theoretical means, but using field theoretical concepts. Thus before trying to axiomatize P , we have to exchange it with a property P 0 that is as close as possible to P , but which is defined by group theo-

ii retical means. We discuss in detail the property of absolute Galois groups stated in the Artin-Schreier Theorem (see Section 3.10), and the property formulated in the so-called “Elementary Type Conjecture” (see Section 3.4). Moreover we discuss whether or not the cyclotomic quotient of an absolute Galois group is first-order definable, see Section 3.3. Chapter 4 is of purely algebraic nature. We discuss so-called Demushkin groups, examples of which are the maximal pro-p-quotients of absolute Galois groups of p-adic fields. First we develop the necessary tools needed in Section 3.4, where Demushkin groups play a major role. Further we analyze the structure of Demushkin groups which are realized as maximal pro-p-quotients of absolute Galois groups. Finally, in Chapter 5, we give a classification of abelian profinite groups up to elementary equivalence, and prove some model theoretic results about categoricity, stability and the structure of saturated models.

Acknowledgements: In the first place, I would like to thank my supervisor Dr. Jochen Koenigsmann for suggesting me this topic and for many and considerable discussions which he managed to arrange even under unfavorable geographic conditions. Without his advice, this dissertation would not have been possible. Secondly, I would like to thank Prof. Martin Ziegler: Without the discussions with him, the last chapter of this thesis would not exist. Moreover, he often offered mathematical first-help when I was hopelessly stuck in the twists and turns of my confused brain. Many others made valuable contributions. In particular, I am grateful to Dr. Markus Junker for his mathematical, but also emotional support. Finally, I want to apologize to all those who suffered from my moaning and my changing moods in the past few years.

iii Contents

Introduction i

1 Preliminaries 1 1.1 Notations ...... 1 1.2 Proﬁnite groups ...... 2 1.3 Cohomology of proﬁnite groups and Brauer groups ...... 7 1.4 Abelian groups ...... 13 1.5 Valuation theory ...... 17

2 Model theory of profinite groups 21 2.1 Many-sorted logics ...... 21 2.2 The complete inverse system of a profinite group ...... 23 2.3 Elementary equivalence of profinite groups ...... 27

2.4 The expressive power of LIS ...... 31

3 Model theory of absolute Galois groups 37 3.1 Model theory of absolute Galois groups ...... 37 3.2 Formalizations of the Artin-Schreier theorem ...... 39 3.3 Searching for the cyclotomic quotient or “dead ends’ death” ... 46 3.4 The Elementary Type Conjecture ...... 52 3.5 Outlook ...... 62

4 Demushkin groups 63 4.1 Basics about Demushkin groups ...... 63 4.2 Demushkin groups as Galois groups ...... 65 4.3 Demushkin groups as semidirect products ...... 73 4.4 An alternative deﬁnition of Demushkin Galois groups ...... 74

5 A classification of proabelian groups 77 5.1 A first classification ...... 78 5.2 Connecting A, A∗ and S(A) ...... 87 5.3 Some model theory of proabelian groups ...... 91 Chapter 1

Preliminaries

1.1 Notations

For simplicity, we will use the symbol A both for the domain of a first order structure (A, . . .) and for the structure A itself. Moreover, we will not distinguish notationally between the symbols of a language L and their interpretation in a given structure. If A is an L-structure for some first-order language L and if C ⊆ A is a parameter set, we denote by AC the expansion of A to the language LC which arises from L by adding new constant symbols for the elements of C. Analogously we define Ac¯ for a tuple c¯ of parameters. If A, B are L-structures and if C is a common subset of A and B, we write AC ≡ BC if the expanded structures are elementarily equivalent. We write Th(A), or ThL(A) if the language L is not clear from the context, for the first-order theory of a structure A. For a formula ϕ, the notation ϕ(x1, . . . , xn) indicates that the free variables of ϕ are among x1, . . . , xn; similarly for a type p(x1, . . . , xn). If the length of the tuple is not of relevance, we sometimes write x¯ for a tuple (x1, . . . , xn). Sometimes we denote with x¯ also the residue class of an element x modulo some equivalence relation; it will be clear from the context which is the at a time intended meaning. Some notations from the algebraic context, which we will use in the sequel without further explanation, are:

P the prime numbers C the complex numbers Fq the ﬁeld with q elements Zp the (group or ring) of p-adic integers idA the identity map on a set A f the restriction of the map f to the set A |A ker(ϕ), im(ϕ) kernel and image of the map ϕ Z/n the cyclic group with n elements ordG(g) or ord(g) the order of the element g in the group G [G : H] the index of the subgroup H in the group G [x, y] the commutator x−1y−1xy of x, y char(K) the characteristic of the ﬁeld K

1 K× the multiplicative group of the field K n ζpn a primitive p -th root of unity (in the current field) µn(K) the set of all n-th roots of unity contained in K. i µp∞ (K) the set of all p -th roots of unity contained in K, i ≥ 1. µp∞ = µp∞ (C) the Prufer¨ group (belonging to the prime p). Gal(L/K) the Galois group of a Galois field extension L/K GK the absolute Galois group of the field K Ka an algebraic closure of the field K Ks a separabel algebraic closure of the field K NL/K the norm map of an algebraic field extension L/K

We will denote both the trivial group and the neutral element of a group with 1, or with 0 in the abelian case.

1.2 Proﬁnite groups

In this section, we brieﬂy recall the facts about proﬁnite groups that we will need. The presentation follows [RZ] and [Wi].

1.2.1 Fundamentals and Conventions

Recall that a profinite group is an inverse limit of an inverse system of finite groups; equivalently, it is a compact, Hausdorff, totally disconnected topological group. A pro-C-group for a suitable class C of finite groups is an inverse limit of groups in C; we are mainly interested in the case that C is the class of all finite groups (leading to arbitrary profinite groups), or of all finite p-groups (leading to pro-p-groups).

If we are dealing with proﬁnite groups and if not otherwise stated, we do always assume a topological background.

This means: Subgroups are assumed to be closed, group homomorphisms to be continuous, isomorphisms to be homeomorphisms etc. Consequently we write U ≤ G for a closed subgroup U of G, and N C G if N is a closed normal subgroup. The symbols ≤o, Co indicate that we are dealing with open (respectively open normal) subgroups; recall that open subgroups of profinite groups are in particular closed. If H is a closed subgroup of G, then H with the subgroup topology inherited from G is profinite; in particular, if (N | N ∈ I) is a family of open normal subgroups of G such that G =∼ lim G/N, then ←−N∈I H =∼ lim H/(H ∩ N). If G ³ H is a continuous epimorphism of profinite ←−N∈I groups, then H is equipped with the quotient topology. A quotient of G modulo Qa closed normal subgroup is called a continuous quotient. The cartesian product Gi of profinite groups Gi is a profinite group when equipped with the product topology. If G is profinite and X is a subset of G, we denote by X its closure in G, and by hXi the closed subgroup generated by X; further h|X|i denotes the smallest closed normal subgroup of G containing X. We say that X generates G if hXi = G;

2 further a set X converges to 1 if every open subgroup of G contains all but a finite number of the elements in X. The rank of G, which we denote by rank(G), is the smallest cardinality of a set of topological generators of G converging to 1. A profinite group G is generated by a countable 1-convergent set if and only if there is a chain G = N0 ≥ N1 ≥ N2 ≥ ... of open normal subgroups of G comprising a base of open neighborhoods of 1. A finitely generated profinite group is determined by its finite continuous images: If two finitely generated profinite groups have the same finite continuous images, then they are isomorphic. An open subgroup of a profinite group has finite index; for finitely generated profinite groups, the converse holds as well. This is not hard to show for pro-p-groups; the general case is a recent and sophisticated result by N. Nikolov and D. Segal, see [NS]. In arbitrary profinite groups, one only knows that open subgroups are closed, and that closed subgroups are open if and only if they have finite index. A profinite group of rank κ has a (free) presentation G = F/N, where F is a free profinite group of rank κ and N is a closed normal subgroup. If G is pro-p, there is also such a presentation where F is free pro-p of rank κ.

1.2.2 The Frattini subgroup

Definition 1.1. For a profinite group G, define the Frattini subgroup Φ(G) to be the intersection of all proper maximal open subgroups of G. For i ≥ 0, define inductively Φ0(G) = G and Φi+1(G) = Φ(Φi(G)); the series Φ(G) ≥ Φ2(G) ≥ ... is called the Frattini series of G. Further for i ≥ 1 let F i(G) := G/Φi(G) be the i-th Frattini quotient of G.

In the sequel, we denote by [G, G] the derived subgroup of G; this is the abstract subgroup of G generated by the commutators. If G is finitely generated, then [G, G] is a closed subgroup. Further Gp is the abstract subgroup of G generated by the p-th powers of G. Lemma 1.2. Let G, H be pro-p-groups. 1) Φ(G) is open if and only if G is finitely generated. 2) Φ(G) = Gp[G, G]. 3) Every continuous epimorphism ϕ : G → H restricts to an epimorphism from Φi(G) to Φi(H) and induces a continuous epimorphism ϕ : F i(G) → F i(H). If G and H are moreover finitely generated, we further have 4) All Φi(G) are open normal subgroups of G. 5) Φ(G) = Gp[G, G]. 6) The Frattini series constitutes a base of open neighborhoods of 1 . Thus if G has the same Frattini quotients as H, then G and H are isomorphic. 7) G/Φ(G) =∼ (Z/p )rank(G); in particular, [G : Φ(G)] = prank(G). Corollary 1.3. If G is finitely generated and H ≤ G is a closed subgroup, then H has countable rank.

Proof: The intersections H ∩ Φi(G) form a countable base of open neighborhoods of 1 in H; thus H is generated by a countable 1-convergent set, cf. Subsection 1.2.1. ¥

3 Remark 1.4. 1) All Φi(G) are characteristic and thus normal subgroups of G. Thus if G is finitely generated, one can give the following equivalent definition: 2) Φ(G) is the intersectionT of all proper maximal open normal subgroups of G. If i ≥ 1, then Φi+1(G) = N, where N runs through those open normal subgroups of G which are proper maximal subgroups of Φi(G) with this property. Remark 1.5. Note that the profinite version of the Nielsen-Schreier Theorem implies that if G is finitely generated of rank n and if U ≤ G is an open subgroup of index d, then U is finitely generated of rank at most 1+d(n−1), see Corollary 3.6.3 in [RZ]. Thus it follows inductively that if rank(G) is finite, then for every i there exist bounds for the rank of Φi(G) and for the cardinality of F i(G). Lemma 1.6. Let G, H be finitely generated pro-p-groups. Then

F i(G) =∼ F i+1(G)/Φi(F i+1(G)).

Thus if F i+1(G) =∼ F i+1(H), then F i(G) =∼ F i(H).

Proof: The reader easily assures himself that for an open normal subgroup N of G holds Φi(G/N) = Φi(G)N/N, compare with the proof of Proposition 2.8.13 in [RZ]. Thus

F i+1(G)/Φi(F i+1(G)) = (G/Φi+1(G))/Φi(G/Φi+1(G)) = (G/Φi+1(G))/(Φi(G)Φi+1(G)/Φi+1(G)) = (G/Φi+1(G))/(Φi(G)/Φi+1(G)) =∼ G/Φi(G) = F i(G).

Lemma 1.7. Let G = F/R be a free presentation of G. Then

F i(G) =∼ F/Φi(F )R.

Proof: We have

F i(G) = (F/R)/Φi(F/R) = (F/R)/(Φi(F )R/R) =∼ F/Φi(F )R.

1.2.3 Free pro-p products

Deﬁnition 1.8. Let G1,G2 be pro-p-groups. The free pro-p-product of G1 and G2 consists of a pro-p-group G1 ∗p G2 and continuous homomorphisms ϕi : Gi → G1 ∗p G2, such that for any pro-p-group H and any continuous homomorphisms ψi : Gi → H for i = 1, 2 there is a unique continuous homomorphism ψ : G1 ∗p G2 → H such that the diagram

4 G1

ψ1 ϕ1

! G1 ∗p G2 H

ϕ2 ψ2

G2 commutes. Remark 1.9. 1) The pro-p product exists and is unique up to isomorphism. 2) Let G be the abstract free product of G1 and G2. Then G1 ∗p G2 is the completion of G with respect to

−1 {N C G | χi (N) Co Gi for i = 1, 2 and G/N is a ﬁnite p-group},

where χi : Gi → G for i = 1, 2 are the natural embeddings. If both G1 and G2 are ﬁnitely generated, this is just the pro-p-completion of G. 3) The natural homomorphism G → G1 ∗p G2 is a monomorphism.

Lemma 1.10. For i = 1, 2, let Fi be a pro-p-group and Ri a closed normal subgroup of Fi. Let Gi = Fi/Ri. Then ∼ G1 ∗p G2 = (F1 ∗p F2)/h|R1,R2|i.

Proof: The corresponding fact is well known for abstract free products and can for example be found in [Ro]. Let ∗abs denote the abstract free product. Consider the commutative diagram

F1 ∗p F2

ψ F1 ∗abs F2 τ

Fi Gi G1 ∗abs G2 G1 ∗p G2

The kernel of τ is the normal subgroup N of F1 ∗abs F2 generated by R1 and R2, see 6.2.7 in [Ro]. As one easily sees, the kernel of ψ is the closure of N in F1 ∗p F2, thus is the closed normal subgroup h|R1,R2|i generated by R1 and R2. ¥ Analogously to the above one can define free pro-C-products for every variety C of finite groups; in particular, the free profinite product of profinite groups exists. The free profinite product of profinite groups G and H will be denoted by G ∗ H.

5 1.2.4 Pontryagin Duality

It is well known that the construction of the dual group establishes an equivalence between the category LCA of locally compact abelian groups with continuous homomorphisms as morphisms and its opposite category LCAopp. As we are only interested in the subcategory of abelian profinite groups, we confine ourselves to this case. Notations and conventions: As we are dealing with abelian groups, we use additive notation. A proabelian group is an abelian profinite group, thus an inverse limit of finite abelian groups. A discrete abelian torsion group is an abelian torsion group that we consider as equipped with the discrete topology; it is thus locally compact. In particular, we consider Q/Z as a discrete abelian group. We will usually denote abstract abelian groups by A, B,..., while A, B, . . . is reserved for proabelian groups. For an abelian group A, let nA = {na | a ∈ A} and A[n] := {a ∈ A | na = 0}. Note that if A is proabelian, then both nA and A[n] are closed subgroups of A. An abelian group A has finite exponent if there is n ≥ 1 such that nA = 0, or equivalently A[n] = A. Definition 1.11. Let PAG be the category of proabelian groups with continuous homomorphisms as morphisms, and let DAT be the category of discrete abelian torsion groups with continuous (thus arbitrary) homomorphisms as morphisms. For every object A from PAG or DAT, let A∗ := {f : A → Q/Z | f is a continuous homomorphism}; if ϕ : A → B is a morphism either in PAG or in DAT, let ϕ∗ : B∗ → A∗, B∗ 3 f 7→ f ◦ ϕ.

Theorem 1.12 (Pontryagin Duality). The above dual group functor ∗ is a duality between PAG and DAT. We denote both the functor from PAG to DAT and the functor from DAT to PAG by ∗. The double dual is naturally isomorphic to the identity functor via the isomorphism ∗∗ 0 αA : A → A , αA(a) := a , where a0 : A∗ → Q/Z is defined by a0(f) = f(a). Remark 1.13. 1) ∗ is an exact functor. Thus ϕ : A → B is an epimorphism of proabelian groups if and only if ϕ∗ : B∗ → A∗ is a monomorphism of discrete abelian torsion groups. It follows that ∗ restricts to a duality between the category of proabelian groups with continuous epimorphisms as morphisms and the category of discrete abelian torsion groups with continuous monomorphisms as morphisms. 2) If A is a finite abelian group, then A∗ is (non-canonically) isomorphic to A. 3) The dual of a direct product of proabelian groups is the direct sum of the dual groups. The dual of a direct sum of discrete abelian torsion groups is the direct product of the dual groups. In particular, the decomposition of a proabelian group A into the direct product of its p-Sylow groups Ap corresponds to the ∗ 1 ∗ decomposition of A into a direct sum of its p-parts (A )p. 1Cf. the definition on page 14.

6 Let A be an object in PAG or DAT, let B be a closed subgroup of A. Then

∗ AnnA∗ (B) := {f ∈ A | f(b) = 0 for all b ∈ B} is the annihilator of B in A∗. One easily checks that if A is an object in PAG or DAT and N is a closed normal subgroup of A, then the natural embedding ∗ ∗ (A/N) → A has image AnnA∗ (N). Lemma 1.14. Let A be an object in PAG or DAT and let n ∈ N. Then ∗ 1) AnnA∗ (nA) = A [n] and ∗ 2) AnnA∗ (A[n]) = nA . Corollary 1.15. Let A be in PAG or DAT. 1) The image of (A/nA)∗ in A∗ under the natural embedding is A∗[n]; the image of (A/A[n])∗ is A∗/n(A∗). 2) nA = 0 ⇐⇒ n(A∗) = 0.

1.3 Cohomology of proﬁnite groups and Brauer groups

In this section we recapitulate some well-known facts about cohomology of proﬁ- nite groups and about Brauer groups. We keep the presentation short, and assume in particular a basic knowledge of Galois theory. More detailed introductions into the cohomology of proﬁnite groups can be found in [Wi] (very down-to-earth) and in [RZ]. A basic text about Brauer groups is [Ke], a more comprehensive one is [GSZ]. Good introductions into Galois cohomology can be found in [NSW] and in [Se]. All results in this section can be found in at least one of the above sources.

1.3.1 Basic deﬁnitions

Let G be a proﬁnite group and let A be a topological left G-module; this means that A is an abelian topological group and a left G-module, such that the map G × A → A is continuous. Equip Gn with the product topology, and let AG := {a ∈ A | xa = a for all x ∈ G}. For n ≥ 0, the abelian group

Cn(G, A) := {f : Gn → A | f a continuous map} is called the group of (inhomogeneous) n-cochains of G with values in A. We 0 n−1 identify C (G, A) with A. For n ≥ 2, we deﬁne a map ∂n : C (G, A) → Cn(G, A) by

(∂nf)(x1, . . . , xn) := x1f(x2, . . . xn) nX−1 i + (−1) f(x1, . . . xi−1, xixi+1, xi+2, . . . , xn) i=1 n + (−1) f(x1, . . . , xn−1).

7 0 Further, let ∂0 : 0 → C (G, A) = A be the zero map, and deﬁne ∂1 : A → 1 n n C (G, A) by ∂1(a)(x) = xa − a. Let B (G, A) = im(∂n) and Z (G, A) = ker(∂n+1). One easily shows that

Bn(G, A) ⊆ Zn(G, A); thus Hn(G, A) := Zn(G, A)/Bn(G, A) is well deﬁned. We call the elements from Bn(G, A) (inhomogeneous) n-coboundaries and the elements from Zn (G, A) the (inhomogeneous) n-cocycles. The group Hn(G, A) is called the n- th cohomology group of G (with coeﬃcients in A). Example 1.16. 1) We have im(∂0) = 0 and ker(∂1) = {a ∈ A | xa = a for all x ∈ G}, thus H0(G, A) =∼ AG. 2) Further

1 im(∂1) = {f ∈ C (G, A) | there is a ∈ A : f(x) = xa − a for all x ∈ G}

and

1 ker(∂2) = {f ∈ C (G, A) | f(xy) = xf(y) + f(x)}.

If G operates trivially on A, we have im(∂1) = 0 and thus

1 H (G, A) = Homc(G, A),

where Homc(G, A) is the set of continuous homomorphisms from G to A. 3) Let n ≥ 1 and let G be a pro-p-group. Recall that the only possible action of G on Z/p is the trivial action (as orbits of elements have p-power-order). For every f ∈ Homc(G, Z/p ), the kernel ker(f) is a subgroup of index p, hence a maximal subgroup, and thus contains the Frattini subgroup Φ(G) of G. It follows that

1 ∼ H (G, Z/p ) = Homc(G, Z/p ) = Homc(G/Φ(G), Z/p ).

∼ n 1 If G has ﬁnite rank n, then G/Φ(G) = (Z/p ) and dimFp H (G, Z/p ) = n is the minimal number of (topological) generators of G. 4) Let G be a pro-p-group of rank n; let G = F/R, where F is a free pro-p- group of rank n and R is a closed normal subgroup. Then one can show that 2 dimFp H (G, Z/p ) is the minimal number h of elements r1, . . . , rh of R such that R = h|r1, . . . , rh|i is the normal closed subgroup generated by r1, . . . , rn. We call h the relator rank of G.

1.3.2 Compatible pairs of maps

Let θ : G1 → G2 be a continuous group homomorphism and let A1,A2 be a topological G1- respectively G2-module; further let ϕ : A2 → A1 be a continuous group homomorphism. The pair (θ, ϕ) is called compatible if xϕ(a) = ϕ(θ(x)a) for all x ∈ G1 and a ∈ A2. Note that if G1 = G2 = G and θ is the identity map, then the pair is compatible if and only if ϕ is a homomorphism of G-modules.

8 Fact 1.17. Let (θ, ϕ) be a compatible pair of maps like above. Then for each n ∈ N, there is a homomorphism

∗ n n (θ, ϕ) : C (G2,A2) → C (G1,A1), ∗ ((θ, ϕ) (f))(x1, . . . , xn) = ϕ(f(θ(x1), . . . , θ(xn))),

n n which induces a homomorphism H (G2,A2)→H (G1,A1) of cohomology groups. Example 1.18 (Restriction map). Let U ≤ G be a closed subgroup and let j : U,→ G be the inclusion map. Then for every G-module A, the pair (j, idA) is compatible and induces a homomorphism

n n n resG,U : H (G, A) → H (U, A), f + Bn(G, A) 7→ f + Bn(U, A). |U n

1.3.3 Cup products

Let A, B, C be topological G-modules. Fix a bilinear map

◦ : A × B → C; we write a ◦ b for the image of (a, b) in C. Suppose further that ◦ : A × B → C is a G-pairing, i.e. (xa) ◦ (xb) = x(a ◦ b) for all a ∈ A, b ∈ B and x ∈ G. For f ∈ Cr(G, A) and g ∈ Cs(G, B) we deﬁne f◦˜ g ∈ Cr+s(G, C) by the equa- tion

(f◦˜ g)(x1, . . . , xr+s) := f(x1, . . . , xr) ◦ (x1 . . . xrg(xr+1, . . . , xr+s)).

Fact 1.19. Let ◦ : A × B → C be a G-pairing of G-modules. Then the map ◦˜ from above induces a well deﬁned bilinear map

∪ : Hr(G, A) × Hs(G, B) → Hr+s(G, C), the cup product.

Example 1.20. If R is a ring on which G acts trivially, then the multiplication R × R → R is a G-pairing; thus the cup product ∪ : Hr(G, R) × Hs(G, R) → Hr+s(G, R) is well deﬁned and bilinear. In the sequel, we are especially interested in the case A = B = C = Z/p for a prime p, the pairing being the multiplication on Z/p .

1.3.4 Brauer groups

Let K be a ﬁeld. Recall that an Azumaya-algebra over K is a ﬁnite dimensional central simple algebra over K.

9 Remark and Deﬁnition 1.21. The group of equivalence classes

Br(K) := { [A] | A is an Azumaya-algebra over K}, where

[A] = [B] ⇐⇒ there is a skew field D over K and some ∼ ∼ k, n ∈ N such that A = Mn(D) and B = Mk(D) with the multiplication [A] · [B] = [A ⊗K B], is called the Brauer group of K. Br(K) is an abelian group with neutral element [K]; the inverse element to [A] is the equivalence class [Aopp] of the opposite algebra of A. We also write A ∼ B for [A] = [B]. Further let nBr(K) := { [A] ∈ Br(K) | n[A] ∼ K}. Remark and Definition 1.22. 1) If L ⊇ K is a field extension, then the restriction map resL/K : Br(K) → Br(L), [A] 7→ [A ⊗K L] is a well defined group homomorphism. 2) Let Br(L/K) := ker(resL/K ). 3) If [A] ∈ Br(L/K), then L is called a splitting field for [A]; we call L as well a splitting field for A. 4) For every field K and every [A] ∈ Br(K), there is a finite Galois exten- Ssion L/K such that L is a splitting field for [A]. It follows that Br(K) = Br(L/K), where L runs through the finite Galois extensions of K.

Definition 1.23. Let K be a field of characteristic q 6=√ p containing a primitive p p p-th root of unity. Let a ∈ K \ K . Define Ka := K( a); further let Na be the image of Ka under the norm map NKa/K : Ka → K.

Note that if K contains a primitive pi-th root of unity, then for every Galois i extension√ L/K with Galois group Z/p , there exists a ∈ K such that L = K( pi a). In the remainder of this section, we assume that K is a field of characteristic 0. The following results remain valid for fields of characteristic q not dividing the degree of the considered field extension. However we will use them in the sequel only for fields of characteristic 0, and thus confine ourselves to this case. Definition 1.24. Let n ≥ 1, let K be a field of characteristic 0 which contains × a fixed primitive n-th root of unity ζn, and let a, b ∈ K . The norm rest algebra (a, b, ζn) (or (a, b) for short, if the choice of ζn is clear from the context) is a K-algebra with generators u, v subject to relations

n n u = a, v = b, vu = ζnuv.

Fact 1.25. Let n ≥ 1, let K be a ﬁeld of characteristic 0 which contains a × ﬁxed primitive n-th root of unity ζn, and let a, b ∈ K . The norm rest algebra (a, b) = (a, b, ζn) has the following properties:

1) (a, b) is an Azumaya-algebra over K whose order divides n. Both Ka and Kb are splitting ﬁelds for (a, b).

10 2) For all a0, b0 ∈ K×:

0 0 0 0 (a, b) ⊗K (a , b) ∼ (aa , b) and (a, b) ⊗K (a, b ) ∼ (a, bb ).

3) (a, b) ∼ K ⇐⇒ a ∈ Nb ⇐⇒ b ∈ Na. 4) (a, b)opp =∼ (b, a). 5) If L/K is cyclic of degree p, say L = Ka, then the induced map

(a, ): K×/(K×)n → Br(L/K), b 7→ (a, b),

2 has Na as its kernel .

1.3.5 Galois cohomology

Remark and Definition 1.26. For a field K, let K(p) denote the maximal Galois p-extension of K, i.e. the field generated by all finite Galois extensions of p-power degree of K. Note that (K(p))(p) = K(p). Further let GK (p) = Gal(K(p)/K); then GK (p) is the maximal pro-p quotient of GK . An absolute pro-p Galois group is the maximal pro-p-quotient of an absolute Galois group.

Let K be a field of characteristic 0 and let L/K be a Galois extension. We work in some fixed algebraic closure Ka of K. One can easily check that the following operations are continuous and thus define topological G-modules: • The operation of Gal(L/K) on the multiplicative group L× of L. a • For n ≥ 1 the operation of GK on the group µn(K ) of n-th roots of unity in Ka. • For n ≥ 0 and if K contains a primitive p-th root of unity, the operation of a GK (p) on the group µpn (K ) = µpn (K(p)). Fact 1.27 (Kummer Theory). Let K be a field of characteristic 0. Then

× × n ∼ 1 a K /(K ) = H (GK , µn(K )); the isomorphism is induced by the map

× n a (K ) 7→ fa, √ n x( a) a fa(x) = √ ∈ µn(K ) for x ∈ GK . n a

If K contains a primitive p-th root of unity, one gets a corresponding isomorphism × × pn ∼ 1 K /(K ) = H (GK (p), µpn (K(p)).

The following corollary is sometimes part of what is called “Kummer Theory”. As it cannot be found in the references at the beginning of this section, we give a proof.

2The map ( , ): K×/(K×)n × K×/(K×)n → nBr(K) is in general not surjective; but by the Merkurjev-Suslin theorem, the image of K×/(K×)n × K×/(K×)n under the map ( , ) generates nBr(K).

11 Corollary 1.28. Let K be a ﬁeld of characteristic 0. Further let K contain × × n a primitive n-th root of unity ζn. For a ∈ K , let ha · (K ) i be the cyclic subgroup generated by a(K×)n in K×/(K×)n. Then the map

× n ha · (K ) i 7→ Ka/K is a well-deﬁned bijection between the set of cyclic subgroups of order n in K×/(K×)n and the set of cyclic extensions of degree n of K.

Proof: It is clear that the map is well-defined and surjective. Assume that × Ka = Kb for a, b ∈ K . Then ker(fa) = ker(fb) = GKa , where fa, fb are the maps defined in Fact 1.27. Hence fa and fb differ only by an automorphism of a µn(K). Note that GK operates trivially on µn(K ) = µn(K), thus fa and fb are i homomorphisms. Every such automorphism is given by an assignment ζn 7→ ζn for an appropriate i ∈ {1, . . . , n − 1}; the concatenation of fa with this map is i × n × n fai . Thus fb = fai , hence from Fact 1.27, follows that a (K ) = b(K ) , and hence a(K×)n and b(K×)n generate the same subgroups. ¥ Definition 1.29. Let L/K be a Galois extension of degree n with Galois group G, let f + B2(G, L×) ∈ H2(G, L×). Then the n2-dimensional K-vector space M A := Lux (ux formal symbols) x∈G becomes an Azumaya-algebra (L, G, f) over K when equipped with the multiplication Ã !   X X X lxux ·  kyuy = lx x(ky)f(x, y)uxy. x∈G y∈G x,y∈G (L, G, f) is called the crossed product of G and L with f. Fact 1.30. Let K be a field of characteristic 0 and let L/K be a finite Galois extension with Galois group G. Then

H2(G, L×) =∼ Br(L/K) via f + B2(G, L×) 7→ (L, G, f).

Remark 1.31. 1) There is an induced isomorphism 2 a × ∼ H (GK , (K ) ) = Br(K).

2) Under the isomorphism from 1), the restriction resL/K : Br(K) → Br(L) of Brauer groups corresponds to the restriction res2 : H2(G , (K×)a) GK ,GL K 2 × a → H (GL, (K ) ) of cohomology groups. 3) Moreover, there is an induced isomorphism 2 a ∼ H (GK , µn(K )) = nBr(K). 4) If p is a prime and K contains a primitive p-th root of unity, then there is an isomorphism 2 ∼ n H (GK (p), µpn (K(p))) = p Br(K) for every n ∈ N.

12 Fact 1.32. Let K be a ﬁeld of characteristic 0 containing a ﬁxed primitive root of unity ζp. Further let G = GK (p) and µp = µp(K). Then the diagram

× × p × × p ( , ) K /(K ) × K /(K ) pBr(K) = ∼ = ∼ = ∼

1 1 ∪ 2 H (G, µp) × H (G, µp) H (G, µp) , the arrows being the maps from the preceding subsections, commutes. Thereby i j ij the cup product is formed relative to the pairing µp × µp → µp, (ζp, ζp) 7→ ζp .

Remark 1.33. ∼ i 1) Using the isomorphism µp = Z/p , ζp 7→ i, the diagram can we written as × × p × × p ( , ) K /(K ) × K /(K ) pBr(K) = ∼ = ∼ = ∼

∪ H1(G, Z/p) × H1(G, Z/p) H2(G, Z/p) ; the pairing is then just the multiplication Z/p × Z/p → Z/p . × 2 2) Let a, b ∈ K . Then fa,b := fa◦˜fb : G → Z/p (compare the deﬁnition of the cup product in Subsection 1.3.3) is deﬁned by

fa,b(x1, x2) = ij, √ √ x ( p a) x ( p b) where i, j are given by the equations 1 √ = ζi and 2 √ = ζj. By the p a p p b p above, (a, b, ζp) ∼ K ⇐⇒ fa,b is a coboundary, i.e. if and only if there is a continuous map u : G → Z/p such that

fa,b(x1, x2) = u(x2) + u(x1) − u(x1x2).

1.4 Abelian groups

In this section, the necessary algebraic and model theoretic knowledge about abelian groups needed in Chapter 5 is provided. Notations: We will use additive notation, and we will denote abstract abelian groups by A, B,..., while A, B, . . . is reserved for proabelian groups. For an abelian group A, let Tor(A) denote the torsion part of A. If A is an abelian group and α is a cardinal, we denote by A(α) the direct sum of α-many copies of A, and by Aα the direct product of α-many copies of A. All groups in this section are assumed to be abelian, even if not explicitly mentioned.

13 1.4.1 Basics

In this subsection, we recapitulate the basic algebraic facts about abelian groups that will be needed in Chapter 5. Detailed treatments of the subject can be found in [Fu] or [Ka]. If A is an abelian group and if every element of A has order pn for some n ∈ N, then A is a p-group. If n ∈ Z, then n|a (n divides a) if there is b ∈ A such that nb = a. An element a ∈ A has infinite height if for every n ≥ 1, there is b ∈ A such that nb = a. Similarly a ∈ A has infinite p-height if for every n ∈ N, there is b ∈ A such that pnb = a. The abelian group A is divisible if all elements have infinite height; similarly A is p-divisible if all elements have infinite p-height. A group is called reduced if it has no divisible subgroups other than 0. A subgroup B of A is a pure subgroup, written B v A for short, if for every b ∈ B and n ∈ N holds: If there is a ∈ A with na = b, then there is b0 ∈ B with nb0 = b; p-purity is defined analogously.

Every abelian group A can be written in the form A = Ar ⊕ Ad, where Ad is divisible and Ar is reduced. Note that Ar might contain elements of infinite height! The maximal divisible subgroup Ad of A is uniquely determined, whereas Ar is only determined up to isomorphism. The divisible group Ad is a direct sum of copies of Q and Prufer¨ groups Lµp∞ , where p ranges over the set of primes. If A is a torsion group, then Ar = p Ap with p-groups Ap; these are called the p-parts (or p-primary parts) of Ar. If ApLhas finite exponent, it decomposes as a direct sum of cyclic groups. The sum p Ap is of finite exponent if and only if all p-parts Ap have finite exponent, and if additionally only finitely many of them are not equal to 0.

Ulm invariants

Let A be an abelian group and let p be a prime. Deﬁne inductively for every ordinal α a subgroup Gα(A) of A via

G0(A) = A,

Gα+1(A) = pGα(A), \ Gα(A) = Gβ(A) for limit ordinals α. β<α

Further let Pα = Gα(A)[p]. Then Pα/Pα+1 is an Fp-vector space; we write up,α(A) for its Fp-dimension. up,α(A) is called the α-th Ulm invariant of A.

If A is a direct sum of cyclic groups and if n ∈ N, then the Ulm invariant up,n(A) is n+1 the number of cyclic summands of order p in A. Note that for everyL abelian group A, we have up,α(A) = up,α(Tor(A)); if A is torsion and A = p Ap is a decomposition into its p-parts, then up,α(A) = up,α(Ap). If n ∈ N, we have

¡ n n+1 ¢ up,n(A) = dimFp (p A)[p]/(p A)[p] ¡ n n+2 n+1 n+2 ¢ = dimFp (p (A[p m]))[p]/(p (A[p m]))[p] n+2 = up,n(A[p m])

14 for arbitrary m ≥ 1. Notation: We will occasionally write pnA[p] instead of (pnA)[p].

1.4.2 The elementary theory of abelian groups

In [Sz], W. Szmielew gave group theoretic invariants which characterize abelian groups up to elementary equivalence. Since then, the model theory of abelian groups (and more general, of modules) has been studied extensively. All results in this section are well known; one can ﬁnd them in [EkF] or [Z].

Notations: In this subsection, LG = {+, −, 0} denotes the usual language for abelian groups. If α is a cardinal, define ½ α if α is finite lv(α) = ∞ if α is infinite to be the logical value of α.

Elementary invariants for abelian groups

Deﬁnition 1.34. Let A be an abelian group. Let

n n+1 Tfp(A) := lim dimFp (p A/p A), n→∞

n Divp(A) := lim dimF ((p A)[p]), n→∞ p

½ 0 if A has ﬁnite exponent E(A) := ∞ otherwise.

Further let as before (up,n(A))n∈N,p∈P be the Ulm invariants of A. We refer to 3 the invariants up,n(A), Tfp(A), Divp(A) and E(A) as to the EF-invariants of A, and to their logical values lv(up,n(A)), lv(Tfp(A)), lv(Divp(A)) and to E(A) as to the elementary EF-invariants of A. Fact 1.35. Let A, B be abelian groups. Then 1) A ≡ B if and only if A and B have the same elementary EF-invariants. 2) A ≺ B if and only if A is a pure subgroup of B, and A and B have the same elementary EF-invariants.

Fact 1.36. Let (Ai)i∈I be a family of abelian groups. Then Y M Ai ≡ Ai. i∈I i∈I Theorem 1.37. Every abelian group A is elementarily to the direct sum

M M (Tf (A)) M n+1 (up,n(A)) p (Divp(A)) δ (Z/p ) ⊕ Zp ⊕ µp∞ ⊕ Q , n∈N, p prime p prime p prime

3EF stands for Eklof and Fisher.

15 where ½ 0 if Exp(A) = 0 δ = 1 if Exp(A) = ∞ .

For further applications, we record the invariants of some special groups: Lemma 1.38. Let p be a prime.

1) Let A := Z/p n+1. Then:

up,n(A) = 1

up,k(A) = 0 for k 6= n

Tfp(A) = Divp(A) = 0 E(A) = 0

2) Let A := Zp. Then for arbitrary n ∈ N:

Tfp(A) = 1

Divp(A) = up,n(A) = 0 E(A) = ∞

3) Let A := µp∞ . Then for arbitrary n ∈ N:

Divp(A) = 1

Tfp(A) = up,n(A) = 0 E(A) = ∞

4) Let A := Q. Then for arbitrary n ∈ N:

E(A) = ∞

Tfp(A) = Divp(A) = 0

up,n(A) = 0 Fact 1.39. Let A ≤ B ≤ C be abelian groups such that A ≺ C and B v C. Then B ≺ C. Corollary 1.40. If A is an abelian torsion group and A ≺ B, then A ≺ Tor(B) ≺ B.

Proof: This is clear, as Tor(B) is obviously pure in B. ¥

Duality of modules

Duality between left- and right modules over a ring R was introduced by M. Prest in [P1]; at that point, it was a duality between formulas for left R- modules and formulas for right R- modules. I. Herzog developed it in [Hz] in a way that it became a duality between complete theories of left R-modules and complete theories of right R-modules, and in certain cases between the left- and right R-

16 modules themselves. In the following, we’ll give a very brief introduction into the subject (based on the introductional part of [P2]) and cite some results that we will use in Section 5.2.2. As we are only interested in abelian groups (thus in Z-modules), we restrict on that case. Thus we don’t have to distinguish between Z operating from the left or from the right; for the general theory, the distinction between left- and right modules is crucial. Let for an arbitrary abelian group A be A+ := Hom(A, Q/Z) the character group of A. If A is an abelian torsion group, then A+ is equal to the Pontryagin dual A∗ of A and we have A+∗ = A∗∗ =∼ A, and A+ is proabelian. For us the following result is central: Theorem 1.41 (Proposition 1.6 in [PRZ]). Let A be an abelian group and let T be a complete theory of abelian groups. Then there is a complete theory T∗ such that4 A ² T ⇐⇒ A+ ² T∗.

Further holds: Theorem 1.42 (Corollary 1.7 in [PRZ]).

A ≺ A++.

Remark: In [PRZ], the authors prove their results more generally for categories of “abelian structures”; this includes the case that T is a complete theory of a right R-module, and especially that T is a complete theory of a Z-module, i.e. of an abelian group.

1.5 Valuation theory

This survey follows the presentation in the lecture “Arithmetic of ﬁelds” hold in Freiburg in the year 2002/2003 by Dr. Jochen Koenigsmann. A good published source where all results can be found is [EnP].

1.5.1 Basics

Recall that a valuation v of a field K with value group Γ is a surjective map v : K → Γ ∪ {∞}, where (Γ, +, <) is an ordered abelian group, such that for all x, y ∈ K holds: • v(x) = ∞ ⇐⇒ x = 0, • v(x · y) = v(x) + v(y), • v(x + y) ≥ min{v(x), v(y)}. Thereby we define ∞ > γ for all γ ∈ Γ, and addition with the element ∞ is defined in the natural way. We call (K, v) a valued field. The ring Ov := {x ∈ K | v(x) ≥ 0} is the valuation ring of v; it contains a unique maximal ideal

4The theory T∗ can be calculated as follows: The theory T is axiomatized by a set of so-called index formulas. To every index formula ψ one can eﬀectively determine the dual formula ψ∗; then T∗ is axiomatized by the set of duals of the index formulas in T.

17 Mv := {x ∈ K | v(x) > 0}. The quotient Kv := Ov/Mv is the residue ﬁeld of (K, v). For an element x ∈ K, we denote its residue class in Kv by x. If K,L are ﬁelds, then a map ϕ : K → L ∪ {∞} is a place if for all x, y ∈ K holds: • ϕ(x + y) = ϕ(x) + ϕ(y), • ϕ(x · y) = ϕ(x) · ϕ(y), • ϕ(1) = 1.

If (K, v) is a valued ﬁeld, then the map ϕv which is the canonical projection Ov → Kv on Ov, and maps every x ∈ K \Ov to ∞, is a place. If ϕ is a place of K, then O = Oϕ = {x ∈ K | ϕ(x) 6= ∞} is the valuation ring of the valuation × × × vϕ : K → (K /O ), where vϕ is the natural projection on K and vϕ(0) = ∞; × × × × y here K /O is ordered via xO ≤ yO ⇐⇒ x ∈ O.

If (K, v), (L, w) are valued fields with L ⊇ K such that Γv ≤ Γw and w = v, |K we say that (K, v)/(L, w), or L/K for short, is an extension of valued fields. The residue field Kv does then embed into Lw; we usually assume that Kv ⊆ Lw.

1.5.2 The Gauss extension

Let (K, v) be a valued ﬁeld and let x be transcendental over K. Then the valuation w on K(x) deﬁned by

n w(anx + ... + a0) := min{v(ai) | 0 ≤ i ≤ n} is the Gauss extension of v. It is the unique extension of v to K(x) such that w(x) = 0 and such that the residue x¯ of x in K(x)w is transcendental over Kv. We have K(x)w = Kv(¯x) and Γv = Γw.

Lemma 1.43 (Lemma of Gauss). Let (K, v) be a valued ﬁeld, and let f ∈ Ov[X] be a monic polynomial. Further let g, h ∈ K[X] be monic with f = g · h. Then g, h ∈ Ov[X]. In particular, if f has a zero α in K, then α ∈ Ov.

Instead of keeping the value group and extending the residue field (as in the Gauss extension), there is also an extension of (K, v) such that the value group grows and the residue field remains unchanged: Lemma 1.44. Let (K, v) be a valued field and let x be transcendental over K, further let Γ ⊇ Γv be an ordered group. Let γ ∈ Γ be such that nγ 6∈ Γv for n ≥ 1. Then there is a unique extension w of v to K(x) with w(x) = γ. We have K(x)w = Kv and Γw = Γ ⊕ Zγ with the ordering induced from Γ.

1.5.3 The Approximation Theorem

Two valuations v1, v2 of a ﬁeld K are called independent if neither Ov1 ⊆ Ov2 nor Ov2 ⊆ Ov1 .

Theorem 1.45 (Approximation Theorem). Let O1,..., On be the valuation rings and M1,..., Mn the maximal ideals of pairwise independent valuations v1, . . . , vn and let x1 ∈ O1, . . . , xn ∈ On. Then there is x ∈ K such that x−xi ∈ Mi for 1 ≤ i ≤ n.

18 1.5.4 Galuation theory (= Valuation theory of Galois extensions)

Let (L, w)/(K, v) be a Galois extension of valued fields, let G = Gal(L/K). We call e = e(L/K) = [Γw :Γv] the ramification index, and f = f(L/K) = [Lw : Kv] the inertia degree of (L, w)/(K, v), or of w/v for short. We call (L, w)/(K, v), or w/v, immediate if e(L/K) = f(L/K) = 1, and inert if for every n ≥ 1 and every finite subextension L0/K of L/K of degree n holds n = f(L0/K). The extension is (purely) ramified if for every n ≥ 1 and every finite subextension L0/K of L/K of degree n of L/K holds n = e(L0/K), and tame if the residue characteristic does not divide e(L/K). Further we define

• Z = Z(w/v) := {σ ∈ G | for all x ∈ Ow is σ(x)−x ∈ Ow} , the decomposition group of G, • T = T (w/v) := {σ ∈ G | for all x ∈ Ow is σ(x) − x ∈ Mw}, the inertia group of G, • V = V (w/v) := {σ ∈ G | for all x ∈ Ow is σ(x)−x ∈ xMw}, the ramiﬁcation group of G.

The corresponding fixed fields KZ = Fix(Z), KT = Fix(T ) and KV = Fix(V ) are called decomposition field, inertia field and ramification field respectively. We have V ⊆ T ⊆ Z ⊆ G and thus K ⊆ KZ ⊆ KT ⊆ KV ⊆ L. Theorem 1.46 (Theorem Z). Let (L, w)/(K, v) be a Galois extension of valued fields, let G = Gal(L/K). Let wZ = w . Then |Z

1) w is the unique extension of wZ to L. 2) The number of diﬀerent extensions of v to L is [G : Z] = [LZ : K]. 3) All extensions of v to L have isomorphic residue ﬁelds and over Γv isomorphic value groups.

4) wZ /v is immediate, thus (LZ )wZ = Kv and ΓwZ = Γ. Theorem 1.47 (Theorem T ). Let (L, w)/(K, v) be a Galois extension of valued ﬁelds, let G = Gal(L/K). Then the map

Φ: Z → Aut(Lw/Kv), µ ¶ σ : L → L σ 7→ w w , x 7→ σ(x)

s is a continuous group epimorphism with kernel T . We have (LT )wT = Lw ∩ Kv and the extension LT /LZ is inert. Theorem 1.48 (Theorem V ). Let (L, w)/(K, v) be a Galois extension of valued ﬁelds, let G = Gal(L/K). Then the homomorphism

× × Ψ0 : T → Hom(L ,Lw ), Ã ! L× → L× ³w ´ σ 7→ σ(x) , x 7→ x induces a continuous group epimorphism

Ψ: T → Hom(Γw/Γv, µp∞ (Lw))

19 with kernel V . If p = char(Kv), then V is the unique p-Sylow subgroup of T ; thus if char(Kv) = 0, then V = 1. The extension LV /LT is purely ramiﬁed and tame.

1.5.5 Hensel’s Lemma

A valuation v of K is henselian if v has a unique extension to every algebraic extension of K.

Lemma 1.49 (Hensel’s Lemma). For a valued ﬁeld (K, v) are equivalent: 1) (K, v) is henselian. 0 2) For every monic f ∈ Ov[X] and every a ∈ Ov with f(a) = 0 and f (a) 6= 0, there is an element b ∈ Ov with f(b) = 0 and a = b.

Every valued field (K, v) has an up to isomorphism unique henselian extension (Kh, vh), which can be embedded in every henselian extension of (K, v). The extension (Kh, vh) is called henselization of (K, v). The valued field (Kh, vh) is an immediate algebraic extension of (K, v); the henselizations of (K, v) in a fixed algebraic closure are thus just the decomposition fields of Ks/K with regard to arbitrary extensions of v to Ks.

20 Chapter 2

Model theory of proﬁnite groups

This chapter is concerned with the model theory of profinite groups. In Section 2.1 we present some basic facts about many-sorted logics. Section 2.2 introduces the concept of the complete inverse system S(G) of a profinite group G: The many-sorted first-order structure S(G) associated with G determines G up to topological isomorphism. In Section 2.3, we use modified Ehrenfeucht-Fraissé- games to describe elementary equivalence of complete inverse systems. Finally, in Section 2.4, we state first results about axiomatizability.

2.1 Many-sorted logics

Notation: If J is a set of sorts and M is a J-sorted structure, we use Mj for j ∈ J to denote the set of elements of M of sort j. We will sometimes be inaccurate and call both j ∈ J and Mj a sort. In our treatment of profinite groups as first-order structures, we will use a many- sorted language, with one sort for every positive natural number. The language LIS we will use is relational, but sometimes we will add constant symbols for elements from a parameter set. Formulas are built in the usual manner, except that variables have a prescribed sort, and that the scope of a quantifier Qx is restricted to the sort the variable x belongs to. Usually the sorts of a many-sorted structure are assumed to be disjoint. We will drop this assumption. This causes no problem because one could consider a many-sorted structure with disjoint sorts and new relations identifying elements of distinct sorts instead. The relational symbols in a many-sorted language come attached with a tuple of sorts specifying the sorts of the elements in the relation; similar for constant symbols. However, to simplify notation, we will use one symbol without index that stands for a whole family of relations living in different sorts. Strictly speak- ing, we do the following: Let L be a many-sorted language with sorts J. Let for n an L-structure M and a tuple ¯ = (j1, . . . , jn) ∈ J be M¯ = Mj1 × ... × Mjn . n Assume (R¯)¯∈J n is a family of relations, where the ¯ ∈ J are pairwise different S n tuples of sorts, such that if we put R := ¯∈J R¯, then R¯ = R ∩ M¯ for ¯ ∈ J .

21 Then we will use a unique symbol R instead of the sorted symbols R¯. An introduction into many-sorted logics in the classical setting can be found in [C1]. A more detailed explanation of the simpliﬁcations and specialities described above can be found in [C2].

Notation: If (j1, . . . , jn) is a tuple of sorts and x1, . . . , xn are variables of sorts j1, . . . , jn respectively, then the notation ϕ(x1, . . . , xn) for a formula ϕ indicates, as mentioned on page 1, that the free variables of ϕ are among x1, . . . , xn. In contrast, the bounded variables in ϕ might range over arbitrary sorts. We say that ϕ is a formula in the sorts (j1, . . . , jn), and we use the same convention for types. Many classical results (like Compactness, Downward and Upward Löwenheim- Skolem, etc.) remain valid in the many-sorted context. Nevertheless, sometimes slight modifications are needed to transfer results to the many-sorted case. We give some explicit examples, some of them being of further significance, and transfer some definitions to the many-sorted case: • If the number |J| of sorts is infinite, the space of n-types with variables of arbitrary sorts is no longer compact, but still locally compact. The space of n-types in a given n-tuple of sorts remains compact. • A many-sorted structure M is said to be κ-saturated if for all sorts all types in one variable of this sort over arbitrary parameter sets of cardinality < κ are realized in M. • A theory is said to be κ-stable if for all parameter sets A of cardinality ≤ κ and every sort j, there are at most κ-many types of sort j over A. A theory is said to be stable if it is κ-stable for some κ. • If a structure is sort-wise finite, then it is the unique model of its theory. We call such structures small; the concept of smallness replaces the concept of finite structures. • (Ryll-Nardzewski Theorem) A many-sorted theory T is ℵ0-categorical if and only if for every n ∈ N and for every given n-tuple (j1, . . . , jn) of sorts, the space of n-types in the sorts (j1, . . . , jn) is finite. Equivalently, there are up to elementary equivalence only finitely many formulas in the sorts (j1, . . . ,Q jn). • As for one-sorted structures, one can build the ultraproduct i∈I Mi/U of a family of many-sorted structures (Mi)i∈I with respect to an ultrafilter U on I. ÃLos’ theorem remains valid. One can often reduce the many-sorted case to the one-sorted case by replacing the sorts by new relations. The resulting models might have an “unbounded” part of elements not lying in any of the new relations; the “bounded” parts of the one-sorted models correspond to the many-sorted models. One can use this correspondence to prove the classical results for many-sorted logics, see [C1]. For us, being interested in questions of axiomatizability, it is necessary to decide on either the one-sorted or the many-sorted attempt, as axiomatizability depends on this choice: For example, the different scope of the quantifiers in the many-sorted and the one-sorted language affects the definability of subsets of a structure.

22 2.2 The complete inverse system of a proﬁnite group

A profinite group is an abstract group that is additionally equipped with a compact, Hausdorff, totally disconnected group topology. The topological structure is in general not determined by the group structure. Thus using the usual group language does not seem to be the appropriate setting to study the model theory of profinite groups, as it would not cope with the topological nature of the groups. The crucial idea to take into consideration both the algebraic and the topological structure is to axiomatize the inverse system of the profinite group instead of the group itself. This concept was introduced by G. Cherlin, L. van den Dries and A. Macintyre in [CDM]. However, they still use a one-sorted language. The first many-sorted attempt to build up a model theory of profinite groups was done by Z. Chatzidakis in [C1]. The following survey is based on the treatments of the subject in [C1] (which is very detailed) and [C2],[C3] (which are more elegant).

Definition 2.1 (The complete inverse system of a profinite group). Let LIS be the many-sorted language {C³,R,P } with sorts indexed by the positive integers, and where C³ and R are binary and P is a ternary relation symbol 1. For a profinite group G, let S(GS) be the following LIS-structure: The universe of S(G) is . G/N, where N runs through all open normal subgroups of G. Further • gN is of sort n if and only if [G : N] ≤ n. • gN C³ hM : ⇐⇒ N ⊆ M. 2 • R(gN, hM): ⇐⇒ gN C³ hM and gM = hM. • P (g1N1, g2N2, g3N3): ⇐⇒ N1 = N2 = N3 and g1g2N1 = g3N1. S(G) is called the complete inverse system of G.

Remark: We denote the elements of sort k in S(G) by Sk(G) instead of S(G)k. By the above deﬁnition, we have Sk(G) ⊆ Sk+1(G). In the sequel, we will use the notation Sk(G) not only for the set of elements of sort k in S(G), but also for the structure induced on Sk(G) by S(G) . Notation: When we write gN ∈ S(G), we assume that N is an open normal subgroup of G, and that gN is a coset of N.

Remark and Definition 2.2. Let G be a profinite group. Consider the LIS- structure S(G). • C³ is transitive and reflexive; it induces an equivalence relation x ∼ y ⇐⇒ x C³ y and y C³ x; we denote the class of x by [x]. Thus gN ∼ hM if and only if N = M and [gN] = [N] = G/N. • C³ induces a partial ordering J³ on the set S/ ∼ of equivalence classes. Further (S/∼, J³) is a modular lattice, the join of [N] and [M] being [M] ∨

1As mentioned in Section 2.1, this is a simpliﬁed notation. The symbols {C³,R,P } stand for families (C³mn)n,m≥1, (Rmn)n,m≥1, (Pn)n≥1 of sorted relations respectively. Here, the (C³mn)m,n≥1, (Rmn)m,n≥1 for n - m are empty, and it doesn’t matter whether we use symbols (C³mn)m,n≥1, (Rmn)m,n≥1 or (C³mn)n|m, (Rmn)n|m. 2If gN C³ hM, then N C M or equivalently G/N ³ G/M; this explains the notation.

23 [N] = [NM], and the meet being [M] ∧ [N] = [N ∩ M]. We have |[M] ∧ [N]| ≤ |[M]| · |[N]|. • P deﬁnes on every equivalence class a group operation; we write xy = z instead of P (x, y, z). • If x C³ y, then R deﬁnes a group epimorphism πxy :[x] ³ [y]. If gN C³ hM, then the map πNM : G/N ³ G/M is the canonical projection. We denote by [[x]:[y]] the index of ker(πxy) in [x]. Thus if gN C³ hM, then [[N]:[M]] is just [N : M].

The inverse system described by S(G) is called complete as it contains all continuous quotients of G. To describe a profinite group up to isomorphism, any cofinal system of continuous quotients would suffice.

Now we deﬁne an LIS-theory whose models are exactly the complete inverse systems of proﬁnite groups:

Definition 2.3. Let TIS be the following LIS-theory: 1) C³ is transitive and reflexive with a unique maximal element. Let ∼ be the equivalence relation x ∼ y ⇐⇒ x C³ y and y C³ x; we denote the equivalence class of x by [x]. 2) An element x is of sort n if and only if |[x]| ≤ n. S 3 3 3) P ⊆ Sx[x] , and moreover P ∩ [x] defines a group law on [x] for every x. 4) R ⊆ xC³y[x] × [y], and for every x C³ y, the set R ∩ [x] × [y] is the graph of a group homomorphism from [x] to [y] which we denote by πxy. 5) If N is a normal subgroup of [x], then there is a unique [y] with x C³ y such that N = ker(πxy). 6) If x C³ y C³ z, then πxz = πyz ◦ πxy; further πxx = id[x] for all x. 7) The set of ∼- equivalence classes is a lattice with respect to the induced pre- order J³. 3

The theory TIS has the following properties: Remark and Definition 2.4. • Every S ² TIS defines a profinite group G(S), namely the inverse limit over the finite groups {[a] | a ∈ S} with respect to the epimorphisms {πab | a C³ b ∈ S}. • It holds S(G(S)) =∼ S and G(S(G)) =∼ G. • If ϕ : G → H is a continuous epimorphism4 of profinite groups, then

S(ϕ): S(H) → S(G) S(ϕ)(gN) := ϕ−1(gN)

deﬁnes an LIS-embedding. • If S1,S2 ² TIS and f : S1 → S2 is an LIS-embedding, then for every x ∈ S1, ∼ f restricts to an isomorphism [x] = [f(x)]. Thus the maps ψ[x] deﬁned by

3Note that this can be axiomatized using the bound [G :(N ∩ M)] ≤ [G : N] · [G : M]. 4If ϕ is not surjective, then the preimage of a proper coset of im(ϕ) is empty; thus the deﬁnition cannot be extended to arbitrary homomorphisms ϕ : G → H.

24 G(S2)

ψ[x]

=∼ [f(x)] [x]

deﬁne a compatible family {ψ[x] | [x] ∈ S1/ ∼} of epimorphisms; the universal property of the inverse limit guarantees the existence of a continuous epimorphism G(f): G(S2) → G(S1).

One can show that in fact the following holds: Fact 2.5. Let PROFIN denote the category of proﬁnite groups with continuous epimorphisms, and let IS denote the category of models of TIS with LIS- embeddings. Then S : PROFIN → IS and G : IS → PROFIN are contravariant functors. Further S ◦G is naturally isomorphic to the identity functor on IS, and G ◦ S is isomorphic to the identity functor on PROFIN, thus S and G deﬁne a duality of categories.

2.2.1 Substructures

A substructure A of S ² TIS does not need to be a model of TIS itself. In fact, it is a model of TIS if and only if the following two conditions hold: • If a C³ s and a ∈ A, then s ∈ A. • If a, b ∈ A, then there is c ∈ A such that c C³ a and c C³ b. If S = S(G), these two conditions enforce that A is the complete inverse system of a continuous quotient G/N of G.

In the following, by “substructure” we do always mean a substructure that fulﬁlls additionally the above conditions.

We write A b S if A is a substructure of S. We will write S(H) b S(G), meaning that S(H) is a substructure of S(G) belonging to the continuous quotient H of G.

2.2.2 Pro-p-groups

p For every prime p, there is an LIS-theory TIS whose models are exactly the complete inverse systems of pro-p-groups: It suﬃces to add to TIS that every element x of sort k has order pn for some n with pn ≤ k. If G is a pro-p-group, then S1(G) = ... = Sp−1(G), Sp(G) = ... = Sp2−1(G), etc. Therefore if we are dealing with pro-p-groups, we slightly change the deﬁnition of the sorts and let Sk(G) be the set of cosets of open normal subgroups of G of index at most pk, now the sorts starting with a 0-th sort corresponding to G/p0G = G/G. If dealing with pro-p-groups, we will sometimes and without further mentioning use these systems which are adapted to the pro-p-case.

25 2.2.3 Proabelian groups

We saw in Subsection 1.2.4 that Pontryagin duality restricts to a duality between the category of proabelian groups with continuous epimorphisms as morphisms and the category of discrete abelian torsion groups with continuous monomorphisms as morphisms. It follows that the composed functor ∗ ◦ G,

S 7→ G(S)∗, ∗ ∗ ∗ [ϕ : S1 → S2] 7→ [S(ϕ) : G(S1) → G(S2) ] from the category of complete inverse systems of proabelian groups with LIS- embeddings as morphisms and the category of discrete abelian torsion groups with continuous monomorphisms as morphisms is an equivalence of categories. Together with Remark 1.13, we get the following: Let S ² TIS such that G = G(S) is abelian. Further let S(C) b S, where C is a continuous quotient of G. Then we can consider C∗ as a subgroup of G∗; a chain

[Nk] J³ [Nk−1] J³ ... J³ [N1] of equivalence classes in S corresponds to a chain

∗ ∗ ∗ N1 ≤ N2 ≤ ... ≤ Nk

∗ ∗ ∗ of subgroups in G , such that [[Ni+1]:[Ni]] = [Ni+1 : Ni ]. Further [Ni] is ∗ non-canonically isomorphic to Ni . Remark: Note that if A is an inﬁnite proabelian group, then the cardinality of S(A) equals the cardinality of A∗. We ﬁnish this section with an example for a complete inverse system of an abelian pro-p-group:

n Example 2.6. For n ∈ N let Gn = Z/p . We use the pro-p-notation from Subsection 2.2.2. The group Gn has for every 0 ≤ k ≤ n exactly one subgroup k k of index p , namely p Gn. Thus we have ∼ S0(Gn) = Gn/Gn = 0, . ∼ . S1(Gn) = Gn/Gn ∪ Gn/pGn = 0 ∪ Z/p , . . 2 ∼ . . 2 S2(Gn) = Gn/Gn ∪ Gn/pGn ∪ Gn/p Gn = 0 ∪ Z/p ∪ Z/p , ...... n ∼ . . . n Sn(Gn) = Gn/Gn ∪ Gn/pGn ∪ ... ∪ Gn/p Gn = 0 ∪ Z/p ∪ ... ∪ Z/p ; the interpretations of the relations are the obvious ones. Now let U be a free

26 Q ultraﬁlter on N, let S := S(Gn)/U. Then using ÃLos’ theorem one sees that ∼ S0 = 0, ∼ . S1 = 0 ∪ Z/p , ∼ . . 2 S2 = 0 ∪ Z/p ∪ Z/p , . . ∼ . . . i Si = 0 ∪ Z/p ∪ ... ∪ Z/p . . , the interpretations of the relations again being the obvious ones. Thus G(S) is a pro-p-group of rank one (as all continuous quotients have rank 1) with exactly i ∼ one continuous quotient isomorphic to Z/p for every i ∈ N, hence G(S) = Zp.

Remark: Note that a free ultraproduct V of the groups Gn in the usual group language LG is a group LG-equivalent to H = µp∞ ⊕ Zp; this can be easily shown using ÃLos’ Theorem and the EF-invariants introduced in Section 1.4.2. Thus V is not a proﬁnite group: The neutral element of H is p-divisible, hence the neutral element in V is p-divisible as well, and thus V is not proﬁnite.

2.3 Elementary equivalence of proﬁnite groups

The aim of this section is to characterize elementary equivalence of profinite groups by means of a modified Ehrenfeucht-Fraisségame. Recall that we denote the s-th sort of S(G) by Ss(G). Further, if A is a subset of S(G), for example if A = [c] is an equivalence class in S(G), then S(G)A denotes the structure one gets by adding new constants for the elements of A. We start with the following observation: Observation 2.7. Let G, H be profinite groups, let S(C) b S(G),S(H) be a common substructure of S(G) and S(H). Then

S(G)S(C) ≡ S(H)S(C) ⇐⇒ for all s ∈ N, c ∈ Ss(C): Ss(G)[c] ≡ Ss(H)[c]. In particular,

S(G) ≡ S(H) ⇐⇒ for all s ∈ N : Ss(G) ≡ Ss(H).

Proof. Two arbitrary structures are elementarily equivalent if and only if all reducts on finite fragments of the language and on finitely many sorts are equivalent. In our special case, the sorts Ss form an ascending chain of subsets, so we can restrict on reducts on one single but arbitrary sort s and on finite parameter sets contained in this sort. Further, there is no harm in assuming that the parameter set is one single equivalence class [c] ⊆ Ss(C): If

27 parameters c1, . . . , cn ∈ St(C) for some sort t are given, consider the infimum n [c] = [c1]∧...∧[cn] ∈ S(C)/∼ of [c1],..., [cn] with respect to J³. Then |[c]| ≤ t n is finite, thus c is contained in Ss(C) for s = t , and c1, . . . , cn are definable over [c]. Thus if Ss(G)[c] ≡ Ss(H)[c], then St(G){c1,...,cn} ≡ St(H){c1,...,cn}. ¥ Recall that two (one-sorted) structures A, B are elementarily equivalent if and only if for all m ≥ 1, the second player in the Ehrenfeucht-Fraisségame Gm(A, B) has a winning strategy. In the following, if we write that there is a winning strategy for a game, we mean that there is a winning strategy for player II. Thereby game Gm(A, B) has the following rules: m determines the number of moves of the game; in the beginning of the game, both players know the number m. In the i-th move, player I chooses either structure A or structure B and an element ai ∈ A (respectively an element bi ∈ B). Player II chooses an element bi ∈ B (respectively an element ai ∈ A). Player II wins the game if and only if after m moves, the map ai 7→ bi for 1 ≤ i ≤ m is a partial isomorphism from A to B. Our aim is to define a modified game that is better adapted to the structure of our complete inverse systems. The idea is that in every move, the players do not only choose one single element in S(G), but a whole equivalence class [N]; the equivalence class [N] has to be chosen such that [N] J³ [M] for the equivalence class [M] from the previous move, and such that the index [N : M] is bounded. Before defining the exact rules, we first state the following Terminology: In the sequel, when talking about an isomorphism between equivalence classes [N] and [M], we do always mean a group isomorphism. Remark: Let S(G),S(H) be given, let [N], [N 0] ∈ S(G)/∼ and [M], [M 0] ∈ S(H)/∼ such that [N] J³ [N 0] and [M] J³ [M 0]. An isomorphism ϕ :[N] =∼ 0 0 [M] induces an isomorphism ϕ :[N ] → [M ] if and only if ϕ(ker(πNN 0 )) = ker(πMM 0 ). Definition 2.8. Let G, H be profinite groups, let c be a common element of Ss(G) and Ss(H). The classy game between S(G) and S(H) over c with m m,s moves and bound s, which we denote by GIS (S(G)c,S(H)c), has the following rules:

• In the first move, player I chooses an equivalence class [N1] ⊆ Ss2 (G) such that [N1] J³ [c], or an equivalence class [M1] ⊆ Ss2 (H) such that M1 J³ [c]. Player II then chooses [M1] ⊆ Ss2 (H) (respectively [N1] ⊆ Ss2 (G)) and an ∼ isomorphism ϕ1 :[N1] = [M1] such that ϕ1 induces the identity on [c]. • In the i-th move, player I chooses an equivalence class [Ni] ⊆ Ssi+1 (G) such that Ni J³ Ni−1, or an equivalence class [Mi] ⊆ Ssi+1 (H) such that Mi J³ Mi−1. Player II has to choose [Mi] ⊆ Ssi+1 (H) (respectively [Ni] ⊆ Ssi+1 (G)) ∼ and an isomorphism ϕi :[Ni] = [Mi] such that ϕi induces ϕi−1 on [Ni−1]. • Player II wins the game if and only if he can find for all i ≤ m a quotient ∼ [Ni] ⊆ Ssi+1 (G) (respectively [Mi] ⊆ Ssi+1 (H)) and ϕi :[Ni] = [Mi] as required. m,s We write GIS (S(G),S(H)) for the game over the trivial (and definable) equivalence class [G]. Lemma 2.9. Let G, H be profinite groups, let S(C) be a common substructure of S(G) and S(H). Then S(G)S(C) ≡ S(H)S(C) if and only if there are winning m,s strategies for all games GIS (S(G)c,S(H)c), where m, s ≥ 1 and c ∈ Ss(C). In

28 particular, we have S(G) ≡ S(H) if and only if there are winning strategies for m,s the games GIS (S(G),S(H)) for m, s ≥ 1. Proof (See Picture 2.9 on page 30): From Observation 2.7 we know that S(G)S(C) ≡ S(H)S(C) if and only if for all m, s ≥ 1 and all c ∈ Ss(C), the m usual Ehrenfeucht games G (Ss(G)[c],Ss(H)[c]) have winning strategies.

Choose an arbitrary sort s ≥ 1, an arbitrary m ≥ 1 and c ∈ Ss(C). Let W be a m,s winning strategy for the classy game GIS (S(G)c,S(H)c). We show that one can m use W to deduce a winning strategy for the usual game G (Ss(G)[c],Ss(H)[c]).

Assume that in the ﬁrst move, player I chooses an element g1N1 ∈ Ss(G). Let ∼ [N1] = [N1] ∧ [c]. Player II uses W to choose [M1] ⊆ Ss2 (H) and ϕ1 :[N1] = [M1] inducing the identity map on [c]. Then ϕ1(ker(πN1N1 )) is a normal subgroup of [M1]; let [M1] be the unique equivalence class such that ϕ1(ker(πN1N1 )) is the kernel of πM1M1 , cf. axiom 5) of TIS. Then ϕ1 induces an isomorphism ∼ ϕe1 :[N1] = [M1]. Now player II picks the element h1M1 := ϕe1(g1N1) ∈ [M1]. The case that player I starts with h1M1 ∈ Ss(H) is analogous. Note that the isomorphisms ϕ1, ϕe1 and the identity map on [c] are compatible with the projections πN1N1 , πN1c, πM1M1 , πM1c; thus the map g1N1 7→ h1M1 is a partial LIS- isomorphism over the parameter set [c].

Assume now that in the i-th move, player I chooses an element giNi ∈ Ss(G). Let [Ni] := [Ni−1]∧[Ni]. As inductively [Ni−1] ⊆ Ssi (G), we have [Ni] ⊆ Ssi+1 (G). ∼ Player II uses W to choose [Mi] ⊆ Ssi+1 (H) and ϕi :[Ni] = [Mi] inducing ϕi−1 on [Ni−1]. Then ϕ1(ker(πN1Ni )) is a normal subgroup of [Mi]; let [M1] be the unique equivalence class such that ϕ1(ker(πN1Ni )) is the kernel of πMiMi . Now ϕi induces an isomorphism ϕei :[Ni] → [Mi]. Finally, player II chooses in the i-th move the element hiMi := ϕei(giNi) ∈ [Mi]. The case that player I starts with hiMi ∈ Ss(H) is analogous. Note that the isomorphisms ϕ1, . . . , ϕi, ϕe1,..., ϕei and the identity map on [c] are again compatible with the projections; thus the map given by gkNk 7→ hkMk for 1 ≤ k ≤ i is a partial LIS-isomorphism over the parameter set [c]. After m moves, the map gkNk 7→ hkMk for 1 ≤ i ≤ m defines a partial LIS-isomorphism over the parameter set [c]. This shows that from winning strategies for the classy game, one can deduce winning strategies for the usual game. For the other direction, choose again an arbitrary sort s ≥ 1, an arbitrary m ≥ 1 and c ∈ Ss(C). Let W be a winning strategy for the usual game msm+1 G (Ssm+1 (G)[c],Ssm+1 (H)[c]); we show that we can deduce a winning strat- m,s egy for the classy game GIS (Ss(G)c,Ss(H)c). 2 2 If player I chooses in the first move [N1] = {g11, . . . , g1j1 } ⊆ Ss (G), j1 ≤ s , [N1] J³ [c], then player II uses j1-times the strategy W to find successively 2 h11, . . . , h1j1 ∈ Ss (H) corresponding to the g11, . . . g1j1 . Consider the map ϕ1 :

[N1] → {h11, . . . , h1j1 }, g1k 7→ h1k.

2 Claim: {h11, . . . h1j1 } =: [M1] is a full equivalence class in Ss (H), and ϕ1 induces the identity map on [c].

Proof of the claim: Assume there is h ∈ [h11]\{h11, . . . h1j1 }. Then in the next move, player I could choose h; but all elements in [N1] are already spent, thus player II would not be able to answer, and W would not have been a winning strategy. As ϕ1 is a partial LIS-isomorphism over the parameter set [c], it is

29 compatible with the projections πN1c, πM1c and the induced map on [c] is the identity. This proves the claim.

Finally, in the ﬁrst move player II chooses the equivalence class [M1] and the ∼ isomorphism ϕ1 :[N1] = [M1]; analogously if player I starts with [M1] ⊆ Ss2 (H). i+1 i+1 If in the i-th move player I chooses [Ni] = {gi1, . . . giji } ⊆ Ss (G), ji ≤ s , then player II uses W to ﬁnd hi1, . . . hiji corresponding to the gi1, . . . gij1 . Again,

{hi1, . . . hij1 } = [Mi] is a full equivalence class. Note that player II has at this point chosen at most s2+s3+...+si+1 < msm+1 elements, thus the game has not ﬁnished yet. Player II chooses the equivalence class [Mi] and the isomorphism ∼ S ϕi :[Ni] = [Mi], gij 7→ hij. As 1≤k≤i ϕi is a partial LIS-isomorphism over the parameter set [c], it is compatible with the projections between [c], [N1],..., [Ni] and [c], [M1],..., [Mi], and the induced map on [c] is the identity. ∼ After m moves, the maps ϕi :[Ni] = [Mi] for 1 ≤ i ≤ m are as required.

ϕm [Nm−1] ∧ [Nm] = [Nm] [Mm] = [Mm−1] ∧ [Mm]

ϕi [Ni−1] ∧ [Ni] = [Ni] [Mi] = [Mi−1] ∧ [Mi]

ϕei [Ni] [Mi]

ϕ2 [N1] ∧ [N2] = [N2] [M2] = [M1] ∧ [M2]

ϕe2 [N2] [M2]

ϕ1 [c] ∧ [N1] = [N1] [M1] = [c] ∧ [M1]

ϕe1 [N1] [M1]

[c] [c]

Picture 2.9

30 2.4 The expressive power of LIS

In this section we discuss which group theoretic and topological properties of profinite groups can or cannot be axiomatized in LIS. The section is a warm- up for the more difficult problems in Chapter 3, where we discuss properties of absolute Galois groups. We will learn to know some special features of our complete inverse systems, and discover some advantages (but also some disadvantages) of them. Moreover, we compare the expressive power of LIS and of the usual group language LG within the class of profinite groups. Notations and conventions: When we write S(G) ≡ S(H) for complete inverse systems S(G) and S(H), we always mean equivalence of the complete inverse systems in LIS. When we write G ≡ H for profinite groups G and H, we mean equivalence of the groups in the usual group language LG. We will sometimes be inaccurate and identify the categories PROFIN and IS; for example we say “the property of a profinite group to be torsion free is not (LIS)- axiomatizable”, meaning that the property of a complete inverse system S to have torsion free associated group G(S) is not axiomatizable. As usual, we call a class of L-structures elementary if it is the model class of a set of L-sentences; a property P of L-structures is axiomatizable if the class Mod(P ) := {M | ML-structure, P is true in M} is elementary. If K is any class of L-structures, we denote by TK the set of L-sentences being true in all members of K; hence K is elementary if and only if K = Mod(TK). We already know a first and essential result from Section 2.2: The category IS of complete inverse systems, which is dual to the category PROFIN of profinite groups, is axiomatizable in LIS. This is a main reason why LIS seems to be the appropriate language to handle profinite groups, instead of the usual group language LG: An LG-ultraproduct of profinite groups is usually not a profinite group, cf. Example 2.6. Thus the class of profinite groups is not LG-elementary. However, some apparently basic and simple properties are not, or at least not as easy as one would expect, axiomatizable in LIS; thus our inverse systems have some disadvantages as well. For technical reasons, it is easier to find weaknesses in the expressive power of LIS than in the expressive power of LG: To prove that a property P is not axiomatizable in LIS, one can construct an ultraproduct S of complete inverse systems all having the property P , such that P does not hold in S. To prove that P is not axiomatizable in LG restricted to the class of profinite groups, this procedure is mostly useless, as the ultraproduct of profinite groups is not profinite any more. Before we start giving examples, we introduce a useful concept. We use the pro-p notation from Subsection 2.2.2. Lemma 2.10. Let n ≥ 1. Then for every i ∈ N, there is a bound f(i, n) and a Frat formula ϕi,n (x) with x a variable of sort f(i, n), such that for all pro-p-groups G of rank ≤ n the following holds: Frat i 1) S(G) ² ϕi,n (a) ⇐⇒ a ∈ [Φ (G)], and Frat 2) S(G) ² ∃xϕi,n (x).

Remark: The reason to include condition 2) is the following: If G is non-trivial (and thus the i-th Frattini quotients for i > 0 have all cardinality > 1), then

31 . the formula ¬ x = x, with x a variable of sort 0, does always fulfill 1). Thus we include 2) to make sure that the considered sort f(i, n) contains the i-th Frattini quotient. Proof: Recall that [G :Φi(G)] ≤ f(i, n) for an appropriate bound f(i, n), Frat cf. Remark 1.5; thus if ϕi,n (x) is as in 1) with x of sort f(i, n), then S(G) ² Frat ∃xϕi,n (x). Frat . It is clear that there is a formula ϕ0,n (x) as desired: It is just the formula x = x Frat with x a variable of sort 0. Assume that we already have a definition ϕi,n (x) of [Φi(G)]. Now Φi+1(G) is the intersection of all open normal subgroups N of G which are proper subgroups of Φi(G) and which are maximal with this property, see Remark 1.4. Such an open normal subgroup N corresponds to an equivalence class [N] 6= [Φi(G)] with [N] J³ [Φi(G)], and which is maximal with this property; [Φi+1(G)] is the infimum over these equivalence classes. The index of such an open normal subgroup N in G is bounded by [G :Φi+1(G)] ≤ f(i+1, n), i+1 Frat thus [Φ (G)] is definable via the formula ϕi+1,n(x) stating that ¡ ∀y if [y] is maximal with [Φi(G)] 6= [y] J³ [Φi(G)], ¢ then ([x] J³ [y]) , and x is maximal with this property, where x and y are variables of sort f(i + 1, n). ¥

Corollary 2.11. For every n ∈ N, the class of pro-p-groups of rank n is LIS- axiomatizable.

Proof: A pro-p-group G has rank n if and only if |[Φ(G)]| = pn, thus if ¡ Frat n¢ S(G) ² ∃x ϕ1,n (x) ∧ |[x]| = p , where x is a variable of sort n (note that we can choose f(1, n) = pn). ¥ Remark 2.12. Note that for every n ∈ N, the class of profinite groups of rank ≤ n is LIS-axiomatizable: A profinite group G has rank n if and only if all continuous quotients have rank at most n; this is obviously axiomatizable. In contrast, the class of profinite groups of rank equal to n isQ not axiomatizable: Let U be a free ultrafilter on the set of primes, let S(G) := p prime S(Zp)/U. Then all groups Zp have rank 1, but G is the trivial group: For every prime p, all sorts Sk(Zp) for 1 ≤ k < p contain only the trivial equivalence class [Zp]; thus for a fixed k, for almost all primes p the sort Sk(Zp) contains only one element. By ÃLos’ theorem, all Sk(G) contain only one element, thus G is the trivial group.

The first property we consider is commutativity of a profinite group. Commu- tativity of a group is in LG very easily and moreover finitely axiomatizable. In LIS, things are a bit more complicated:

Example 2.13. The class of proabelian groups is LIS-elementary. The class of non-abelian proﬁnite groups is not LIS-elementary, thus the class of proabelian groups is not ﬁnitely axiomatizable.

Proof: A proﬁnite group is abelian if and only if all continuous quotients are abelian. Thus G is abelian if and only if S(G) satisﬁes all formulas . ψn = ∀x∀yxy = yx,

32 for n ≥ 1, where x, y are variables of sort n. Recall that for a finitely generated pro-p-group G, the Frattini subgroups form a fundamental system of open neighborhoods. Thus a finitely generated groups G is abelian if and only if all iterated Frattini quotients are abelian. We use this to prove that the class of proabelian groups is not finitely axiomatizable.

For i ≥ 1, let Zp oχi Zp be defined as in Definitions 3.34 and 3.36. Then for l ∼ i i every l ≥ 1 and for all i > l, we have Gi/Φ (Gi) = Z/p × Z/p , see Corollary 3.42; thus infinitely many of the Gi fulfill

l . S(Gi) ² ∀x∀y ∈ [Φ (Gi)] xy = yx, where x and y are variablesQ of an appropriate sort. Thus if U is a free ultra ﬁlter on N \{0} and S := S(Gi)/U, then

l . S ² ∀x∀y ∈ [Φ (Gi)] xy = yx for all l ≥ 1, thus G(S) is abelian5. Thus the property “being not abelian” is not LIS-axiomatizable, which implies that the property of being abelian is not finitely axiomatizable. ¥ Remark: If we had chosen a one-sorted language instead of the many-sorted language LIS to describe the complete inverse systems of profinite groups, cf. page 22, then the property of being non-abelian would obviously be axiomatizable. We give another example of an apparently simple property which is trivial to axiomatize in LG, but which is not an elementary statement in LIS: Example 2.14. Let p be a prime. Then the class of profinite groups containing an element of order p is not LIS-axiomatizable.

∼ n Proof: Consider Example 2.6 again. All the groups Gn = Z/p contain an Q ∼ element of order p, but the ultraproduct Gn/U = Zp does not. ¥ Let us record the following consequence (which is admittedly not very surprising; the corresponding fact holds for LG as well):

Corollary 2.15. The class of torsion proﬁnite groups is not LIS-elementary.

What about torsion free proﬁnite groups? First we need a characterization of complete inverse systems of torsion free groups. We have Lemma 2.16. Let y ∈ S(G), let [y] = G/N for an open normal subgroup of G. Then the following are equivalent:

1) There is g ∈ G such that y = gN and ordG(g) = ord[N](y). 2) For all open normal subgroups M C N of G, there exists z ∈ [M] with πMN (z) = y and ord[M](z) = ord[N](y).

Proof: Assume that g is as in 1). Then for every M C N, choose z = gM. For the other implication, assume that for every open normal subgroup M C N of G there exists z ∈ [M] with πMN (z) = y and ord(z) = ord(y). For M C N let ZM := {z ∈ [M] | πMN (z) = y and ord(z) = ord(y)}. Then the (ZM )MCN

5In fact, it is easy to see that G(S) is a proabelian torsion free group of rank 2 and thus isomorphic to Zp × Zp.

33 with the restrictions of the projection maps πMN form an inverse system of Hausdorff, compact and non-empty topological spaces; thus the inverse limit Z := lim(Z , π | M C N) is non-empty. Every x ∈ Z defines an element ←− M MN g ∈ G with gN = y and with ord(g) = ord(y), which finishes the proof. ¥ Corollary 2.17. Let G be a profinite group. Then G is torsion free if and only if for all y ∈ S(G), where [y] = G/N for an open normal subgroup N of G, there is an open normal subgroup M C N of G such that ord[M](z) > ord[N](y) for all z ∈ [M] with πMN (z) = y.

Proof: If G is torsion free, then obviously for all y ∈ S(G), [y] = G/N there is no g ∈ G such that y = gN and ord(g) = ord(y). Thus for all y ∈ S(G), [y] = G/N, there is an open normal subgroup M C N of G such that ord(z) > ord(y) for all z ∈ [M] with πMN (z) = y. For the other direction, assume that G is not torsion free, for instance let g ∈ G have order n. Let N be an open normal subgroup of G such that ord(gN) = n; then for all open normal subgroups M C N of G there is z = gM ∈ [M] with πMN (z) = y and ord(z) = ord(y). ¥ As there is no reason why there should be a bound for the cardinality [G : M] in the above corollary, one would conjecture that the class of torsion free profinite groups is not elementary in LIS (unlike the situation in LG!). We will prove this in Chapter 3, Proposition 3.16, with the help of field theoretic arguments. After these a bit disillusioning examples, here is a more encouraging one; others will follow in the next chapter. Recall that a profinite group G is projective if and only if for all diagrams

! ϕ

B A ψ with profinite groups B,A and continuous epimorphisms ϕ, ψ there is a continuous homomorphism χ : G → B which makes the triangle commute6. One can show that it suffices to consider finite groups A and B; further we can identify A with a continuous quotient G/N of G. Next one can split up every map χ like defined above into an epimorphism onto a continuous quotient G/M of G first and then a monomorphism of this quotient into B. Finally, the natural projections of G onto G/N, G/M factorize through G/(N ∩ M), and H := N ∩M is an open normal subgroup of index at most [G : M]·[G : N] of G. Thus we get the following equivalent condition on S(G) for G to be projective:

6An equivalent and more standard deﬁnition is: G is projective if and only if every short exact sequence 1 → N → E → G → 1 of proﬁnite groups splits.

34 For all n ≥ 1, for all equivalence classes [N] of cardinality at most n, for all ﬁnite groups B of cardinality at most n, and for all (∗) epimorphisms ϕ : B → [N], there exists an equivalence class [H] of size at most n2 and a monomorphism b :[H] → B such that ϕ(b(h)) = πHN (h) for all h ∈ [H].

[H]

πHN b

B [N] ϕ

Proposition 2.18. The class of projective proﬁnite groups is LIS-axiomatizable.

Proof: This is clear from the above if we can show that, unless G is trivial, arbitrary finite groups can be coded in S(G). Let B = {b1, . . . , bn} be a finite group, and let [a] be an arbitrary nontrivial equivalence class in S(G). Denote by 1 the neutral element of [a], and fix some n element 1 6= g ∈ [a]. Identify bi with the n-tuple ti ∈ {1 , g} whose i-th component is g and whose other components are 1. The group multiplication on 3 T = {t1, . . . , tn} is determined by the finite and thus definable subset R ⊆ T which is defined by (ti, tj, tk) ∈ R ⇐⇒ bibj = bk. It is a definable property of an arbitrary subset of T 3 to define a group multiplication on T ; thus we can quantify over all group structures on T , and thus axiomatize the statement in (∗). ¥ We don’t know whether it is possible to axiomatize the class of projective profinite groups in LG, but one would reckon that it is not. Observation 2.19. If A is finitely generated and S(A) ≡ S(B) for some profinite group B, then A =∼ B.

Proof: If S(A) ≡ S(B), then A and B have the same continuous images. If moreover A is ﬁnitely generated, then A and B are isomorphic, cf. page 3. ¥

Finally we give examples for classes in which the expressive power of LG and LIS coincide. Proposition 2.20. 1) If A, B are abelian proﬁnite groups, then

A ≡ B ⇐⇒ S(A) ≡ S(B).

2) If A, B are ﬁnitely generated proﬁnite groups, then

A ≡ B ⇐⇒ S(A) ≡ S(B) ⇐⇒ A =∼ B.

35 Proof: 1) is the content of Theorem 5.22 and Theorem 5.26. To prove 2), let A, B be ﬁnitely generated proﬁnite groups. It is shown in [JL] that A ≡ B ⇐⇒ A =∼ B. Now the result follows from the last observation. ¥ Remark:

1) Note that both in 1) and 2), the map that assigns to the LIS-theory ThLIS

(S(A)) of S(A) its LG-theory ThLG (A) is not a homeomorphism of the corresponding Stone spaces: • In the case of proabelian groups, let T1 be the space of LIS-theories of proabelian groups, and let T2 be the space of LG-theories of proabelian groups with the topology inherited from the Stone topology on the space of LG-theories of arbitrary abelian groups. Then T1 is a compact space, whereas T2 is not: Consider for n ≥ 1 the LG-formulas ϕn = (∃x x 6= 0 ∧ ∀x ∃y ny = x), which hold in a group G if and only if G is non-trivial, and if all elements in G are divisible. Then there is no theory T in T2 containing all ϕn: Any model G of T would be non-trivial and contain only divisible elements; but then G would not be profinite (as profinite groups are compact and thus don’t contain divisible elements). Thus the intersection of all the closed sets belonging to the ϕn is empty, but every finite intersection

is not: If ϕn1 , . . . , ϕnk are given, choose a prime p > max{n1, . . . , nk}; then

Z/pZ is a profinite model of ϕn1 ∧ ... ∧ ϕnk . • For finitely generated profinite groups, let T1 be the space of LIS-theories of finitely generated profinite groups and let T2 be the space of LG-theories of finitely generated profinite groups, now T2 equipped with the topology inherited from the space of all LG-theories of groups. Consider the LG- formula ϕ = ¬∀x∀y xy = yx. Then the set U of preimages in T1 of the basic open set corresponding to ϕ in T2 is not open: If U had the form ψ for some LIS-formula ψ, then this formula would axiomatize the non-abelian groups within the groups of finite rank; but such a formula does not exist, cf. Example 2.13. 2) We are still seeking an example for two profinite groups G, H such that G and H are LG-equivalent, but such that S(G),S(H) are not LIS-equivalent. Such an example would in some sense prove that the language LIS is the right one for considering profinite groups as model-theoretic objects. To find such an example one would -as a first trial- start with two groups G, H that are isomorphic as abstract groups and thus are LG-equivalent, but not isomorphic as profinite groups; and then hope that S(G),S(H) are not even elementarily equivalent. Unfortunately, the groups G, H would have to be sufficiently complicated: According to the last example, both finitely generated profinite groups and proabelian groups can’t serve as examples. In finitely generated profinite groups, the abstract group structure already defines the topological structure, and one can not even find G, H that are isomorphic as abstract, but not as topological groups. In contrast, one can at least find proabelian groups G, H that are isomorphic as abstract groups, but not as topological groups, cf. Observation 5.30; but according to the last proposition, S(G) and S(H) will be LIS-equivalent.

36 Chapter 3

Model theory of absolute Galois groups

All classes of objects we consider are assumed to be closed under isomorphisms. Deﬁnition 3.1. Let AG be the class of complete inverse systems of absolute Galois groups and let TAG denote its elementary theory (that is, the set of all LIS-formulas valid in all members of AG).

As already mentioned in the introduction, there is not even a conjectural characterization of the class of absolute Galois groups by group theoretical means. It seems to be likely that AG is not first-order axiomatizable. Thus it is natural to concentrate on the following two points: 1) An isomorphism closed class of L-structures is elementary if and only if it is closed under ultraproducts and under elementary substructures. One already knows that AG is closed under ultraproducts, see Lemma 3.2. Thus to show that AG is not elementary, it would be sufficient to construct an elementary substructure of (the complete inverse system of) an absolute Galois group that is not (the complete inverse system of) an absolute Galois group itself. 2) What are the models of TAG, i.e. what is the smallest elementary class of (complete inverse systems of) profinite groups containing all (complete inverse systems of) absolute Galois groups? To determine this class, we want to find LIS-formulas in TAG which do not follow from TIS. In Section 3.1, we will collect basic facts about the model theory of absolute Galois groups. In Sections 3.2, 3.3 and 3.4, we will discuss in detail the axiomatizability of respectively one property of absolute Galois groups.

3.1 Model theory of absolute Galois groups

In this section, we collect results about complete inverse systems of absolute Galois groups. Those which are not proven here can be found either in [CDM], or in [C3], or are trivial.

37 Let K be a field and let GK be its absolute Galois group. Then the open normal subgroups of GK are exactly the absolute Galois groups GL for finite Galois extensions L of K; thus [ . S(GK ) = Gal(L/K), where L runs through the finite Galois extensions of K. If N = GL and M = GF for some finite Galois extensions F ⊇ L of K, then the projection πMN corresponds to the natural restriction map Gal(F/K) → Gal(L/K). If L/K is a regular extension, the restriction map ϕ : GL → GK is a continuous epimorphism, thus S(ϕ): S(GK ) → S(GL) is an LIS-embedding. In this case, we consider S(GK ) as a substructure of S(GL). As already mentioned before, the class AG is closed under ultraproducts:

Lemma 3.2 (Lemma 19 in [CDMQ]). Let (Ki)i∈I be a family of fields, let U be an ultrafilter on I, and let K := i∈I Ki/U. Then Y ∼ S(GKi )/U = S(GK ). i∈I Remark 3.3. It follows that the class of all absolute Galois groups is elementary if and only if for every field K and every complete inverse system S ≺ S(GK ), ∼ there is a field L such that S = S(GL).

Deﬁnition 3.4. For a prime p, let AGp be the class of complete inverse systems of maximal pro-p quotients of absolute Galois groups.

We consider S(GK (p)) as a substructure of S(GK ). It is easily checked that if (Gi)i∈I are proﬁnite groups and if Gi(p) is the maximal pro-p-quotient of Gi, then ¡ Y ¢ ¡ Y ¢ ∼ G S(Gi)/U (p) = G S(Gi(p))/U ; i∈I i∈I in particular we get:

ObservationQ 3.5. Let (Ki)i∈I be a family of ﬁelds, let U be an ultraﬁlter on I, and let K := i∈I Ki/U. Then Y ∼ S(GKi (p))/U = S(GK (p)).

Deﬁnition 3.6. For n ∈ N, let AG ≤n := {S ∈ AG | rk (G(S)) ≤ n} and let =n AGp := {S ∈ AGp | rk (G(S)) = n}.

≤n =n Corollary 3.7. Let n ∈ N. Then AG and AGp are axiomatizable.

Proof: It suffices to show that the classes are closed under elementary equivalence and under ultraproducts. As finitely generated profinite groups are determined by their LIS-theory up to isomorphism (see Observation 2.19), both classes are closed under elementary equivalence. An ultraproduct of complete inverse systems of pro-p-groups of rank n has again rank n (this follows from Corollary 2.11), and an ultraproduct of complete inverse systems of arbitrary =n profinite groups of rank ≤ n has again rank ≤ n (by Remark 2.12). Thus AGp and AG ≤n are elementary. ¥

38 Note that the above argument does not give an effective procedure to find an axiomatization. In Subsection 3.4.3, we will give a more or less tangible axiomati- =n zation of AGp under the additional assumption that the so-called “Elementary Type Conjecture” (cf. Conjecture 3.40) holds. The following result is due to Cherlin, van den Dries and Macintyre; a proof can be found in [C3]. It states that the theory of S(GK ) is determined by the theory of K: Theorem 3.8 (Lemma 17 in [CDM]). Let K and L be fields.

1) If K ≡ L, then S(GK ) ≡ S(GL). 2) If K is κ-saturated, then so is S(GK ).

In the remaining part of this chapter, we want to find LIS-formulas following from TAG which are not already following from TIS. For this purpose, we will analyze several properties of absolute Galois groups and try to find an LIS- axiomatization of them. The difficulty of this procedure is that a property P (which is defined for and holds in absolute Galois groups) that we check might not be defined by group theoretical means, but using field theoretical concepts. Thus we first have to “translate” the property P into a property P 0 of arbitrary profinite groups. The aim is to find P 0 in such a way that it holds in all absolute Galois groups, but not in all profinite groups. We will of course try to choose P 0 as close as possible to the original property P .

3.2 Formalizations of the Artin-Schreier theorem

In this subsection, we discuss possibilities to axiomatize the property of absolute Galois groups stated in the following theorem:

Theorem 3.9 (Artin-Schreier). Let K be a ﬁeld. If GK is ﬁnite and non-trivial, ∼ then K is real closed and GK = Z/2.

More exactly, the situation is as follows: If K is not formally real, then GK is ∼ torsion free. If K is formally real, then GK = H o Z/2 for a torsion free group H. It is an open problem whether the above property can be axiomatized; we consider the slightly weaker property

All elements of ﬁnite order in G have order 1 or 2. The product (AS) of two distinct elements of order 2 has inﬁnite order.

Theorem 3.10. There is a set ΦAS of LIS-sentences such that for all proﬁnite groups G the following holds:

1) If S(G) ² ΦAS, then G has the property (AS). 2) S(GK ) ² ΦAS for every absolute Galois group GK .

Before we prove Theorem 3.10, we need some technical lemmata. First note the following well-known result, which allows us to restrict in the proof of 3.10.2) on absolute Galois groups of ﬁelds of characteristic 0:

39 Theorem 3.11 (Corollary 22.2.3 in [Ef4]; see also the proof of Proposition 5.1 in [K8]). Every absolute Galois group can be realized over a ﬁeld of characteristic 0.

The problem when one tries to axiomatize (AS) is the following: We know from Corollary 2.17 that a profinite group G is torsion free if and only if for all y ∈ S(G) with [y] = G/N for an open normal subgroup N of G, there is an open normal subgroup M C N of G such that we have ord(z) > ord(y) for all z ∈ [M] with πMN (z) = y. But as mentioned before, in the general context there is no bound for the cardinality of [G : M]. The solution of this problem is that in absolute Galois groups and for those elements we have to consider, there is such a bound. To prove this, one only needs basic Galois theory. However the proofs are technical and lengthy, therefore we will formulate the central Galois theoretic results in some preliminary lemmata. As we are not interested in finding the best possible bounds, we won’t make a sport out of sharpening the bounds in the following proofs; one could easily improve them with some extra arguments. We assume all algebraic field extensions of the current field K to be embedded in a fixed algebraic closure Ka of K. We assume a basic knowledge about Galois theory (Kummer Theory, norm maps etc.), and we will repeatedly use the following well-known fact: Fact 3.12 (Chapter VI, Theorem 9.1 in [La]). Let K be a field and let n ≥ 2. Further let 0 6= a ∈ K. For all p ∈ P such that p | n let a 6∈ Kp, and if 4 | n let additionally a 6∈ −4K4. Then Xn − a is irreducible in K[X]. Lemma 3.13. Let K be a field of characteristic 0, and let L/K be a Galois extension of degree n. Let p 6= 2 be a prime and let σ ∈ Gal(L/K) an element 2 of order p. Further assume that a fixed primitive p -th root of unity ζp2 lies in L. Then there is M ⊇ L with M/K Galois of degree ≤ (pn)! such that all extensions of σ to M have order > p.

Proof (See Picture 3.13 on page 41): Let F denote the fixed field of σ. Let p ζp = (ζp2 ) . We have F (ζp) ⊆ L and [F (ζp): F ] | p − 1; as [√L : F ] = p, it follows p that ζp ∈ F . Using Kummer Theory, we√ can write L√= F ( a) for some a ∈ F , p p p2 a√6∈ F and for some fixed p-th root a of a.√ Let a be a fixed p-th root of p a. Using Fact 3.12, we see that M 0 := L( p2 a) has degree p2 over F , thus √ √ √ 2 √ 0 p2 p2 p2 p −1 p2 [M : L] = p. The conjugates of a over F are a, ζp2 a, . . . , ζp2 a; as 0 ζp2 ∈ L, the extension M /F is normal and has degree np over K. √ √ √ p p p2 By choosing√ ζp appropriately, we can assume that σ( a) = ζp a√. As a and√ p2 p2 p2 ζp2 a are conjugate over F ,√ we can extend√ the partial√ map a 7→√ζp2 a 0 p p2 p p2 p p2 p to√τ ∈ Gal(√M /F ). Then τ( a) = τ(( a) ) = (τ( a)) = (ζp2 a) = p p ζp a = σ( a), thus τ extends σ. Claim: ord(τ) = p2. 1 Proof of the Claim: This is clear if τ(ζp2 ) = ζp2 , as then √ √ √ √ √ √ p p2 p−1 p2 p−1 p2 p p2 p2 p2 τ ( a) = τ (ζp2 a) = ζp2 τ ( a) = ... = ζp2 a = ζp a 6= a,

1 If τ(ζp2 ) = ζp2 it follows that ζp2 ∈ F and we are in the situation of usual Kummer Theory.

40 thus τ has order > p and thus order p2. If τ(ζp2 ) 6= ζp2 , we have ζp2 6∈ F , hence L = F (ζp2 ), and τ generates the |L Galois group of L over F . We get √ √ √ p p2 p−1 p2 p−1 p−1 p2 τ ( a) = τ (ζ 2 a) = τ (ζ 2 )τ ( a) = ... ¡ Q p ¢ √ p √ √ p−1 i p2 p2 p p2 = τ (ζ 2 ) a = N (ζ 2 ) a = (−1) (−ζ ) a √i=0 p L/F p p p2 = ζp a, √ √ so again τ p( p2 a) 6= p2 a. This proves the claim. This shows that Gal(M 0/F ) =∼ Z/p 2. Under the restriction map Gal(M 0/F ) ³ Gal(L/F ) =∼ Z/p , all preimages of elements of order p have order p2; hence all preimages σe of σ in Gal(M 0/F ) have order p2. Now consider the splitting ﬁeld M of M 0 over K. It has degree at most [M 0 : K]! = (pn)! over K, and all extensions of σ to M have order at least p2.

n √ √ p 0 p2 K F = Fix(σ) p L = F ( a) M = F ( a) M 3 3

a ζp2 Picture 3.13

¥ The next lemma states the corresponding fact for an automorphism σ of order 2. In this case, we need the additional assumption that the fixed field of σ does contain a primitive 4-th root of unity i to ensure that σ does not correspond to the “complex conjugation”, which cannot be lifted to an element of order 4. Besides, the proof is similar to the one of the last lemma. Lemma 3.14. Let K be a field of characteristic 0, and let L/K be a Galois extension of degree n. Let σ ∈ Gal(L/K) be an element of order 2. Denote by F the fixed field of σ, and assume additionally that a primitive 4-th root of unity i lies in F . Then there is M ⊇ L with M/K Galois of degree ≤ (2n)! such that all extensions of σ to M have order > 2.

√ 2 Proof:√ Write L = F ( a) for some a ∈ F , a 6∈ F and some fixed square√ root 4 2 4 2 4 0 4 a of a. Then −4F √= (2i) F ⊆ F√ , thus √a 6∈ −4F . Let M := L( a) for some fixed 4-th root 4 a of a with ( 4 a)2 = a; using Fact 3.12 and Kummer Theory, it follows that [M 0 : F ] is a cyclic extension of degree 4. As in Lemma 3.13 we get that all extensions of σ to M 0 have order ≥ 4. The degree of the splitting field M of M 0 over K is at most (2n)! and all extensions of σ to M have order > 2. ¥ Lemma 3.15. Let K be a field of characteristic 0, and let L/K be a Galois extension of degree n. Let σ ∈ Gal(L/K) be an element of order 4. Further assume that L contains a fixed primitive 8-th root of unity ζ8. Then there is M ⊇ L with M/K Galois of degree ≤ (4n)! such that all extensions of σ to M have order at least 8.

41 2 Proof: Let F denote the ﬁxed ﬁeld of σ. Let i := ζ8 . We distinguish the cases i ∈ F and i 6∈ F .

Case 1: i ∈ F . √ 4 2 In this case,√ we have L =√ F ( a√) for some a ∈ F , a 6∈ F and some fixed 4 4 2 4 2 4 2 4-th root a of a with ( a) = a. Then √a 6∈ −4F = (2i) F ⊆ F ; thus√ X8 − a is irreducible over F . Let M 0 := L( 8 a) for some fixed 8-th root 8 a of a, and let M be the splitting field of M over K; one proceeds as in Lemma 3.13 to show that all extensions of σ to M have order at least 8, and that [M : K] ≤ (2n)! < (4n)!. Case 2 (See Picture 3.15): i 6∈ F .

Claim: F (i) = F (ζ8) has degree 2 over F . Proof of the Claim: Assume not; then F (ζ8)/F√ has degree 4√and as ζ8 ∈ L, this implies that L = F (ζ8). It√ follows that 2 6∈ F (i) and 2 ∈ L, as we have ζ = ± 1√±i . Thus L = F (i, 2) and Gal(L/K) ∼ (Z/2)2, a contradic- 8 2 = tion to σ having order 4. This proves the claim. √ This shows that [F (ζ8): F ] = 2 and that √L = F (i, a) for some a ∈ F (i), a 6∈ F (i)2 and for some ﬁxed square root a of a. Moreover, we have a 6∈ −4F (i)4 = (2i)2F (i)4 ⊆ F (i)2; thus X4 − a is irreducible over F (i). Further- more a 6∈ F , as otherwise we would have Gal(L/F ) =∼ (Z/2)2. M

M 0

√ p L( 4 a) L( 4 σ(a))

2 2 √ L = F (i, a)

F (i) 3 a, σ(a)

n 2

F = Fix(σ)

Picture 3.15 √ √ √ p 4 4 2 Fix some 4-th rootp a of a with ( a)p= a, a squarep root σ(a) of σ(a) and a 4-th root 4 σ(a) of σ(a) with ( 4 σ(a))2 = σ(a). The conjugates of √ √ p a over F are the elements of A0 := {± a, ± σ(a)} ⊆ L. The conjugates of

42 √ √ √ p p 4 a over F are the elements of B := {± 4 a, ±i 4 a, ± 4 σ(a), ±i 4 σ(a)}. Let √ p 0 0 4 4 0 0 M := L( a, σ(a)); as i ∈ L, we have B0 ⊆ M and thus M /F is normal of degree at most 4n. √ How does σ operate√ on A√0? As a 6∈ F (i), the restrictionp of σ to A0 has order 4; thus σ( a) 6= ± a. By choosing the square root σ(a) properly (i.e. choosing the one with the correct signum), we can assume that √ p √ p a 7→σ σ(a) 7→σ − a 7→σ − σ(a). (∗) Now consider an arbitrary extension τ of σ to M 0. Claim: τ has order at least 8 (and thus equal to 8). Proof of the claim: √ √ √ √ 4 4 4 4 • We know that τ( a) ∈ B0. If τ( a) ∈ {± a, ±i √a}, then pτ(a) = σ(a) = a; a contradiction, as a 6∈ F . Analogously, if τ( 4 a) = ±i 4 σ(a), √ p √ p 4 4 then τ( a) = − σ(pa), a contradictionp to (∗). Thus pτ( a) = ±p σ(a). 4 4 4 4 • We also know that τ( σ(a)) ∈ B0. If τ( σ(a)) ∈ {± σ(a), ±pi σ(a)}, then τ(σ(a)) = σ2(a) = σ(a); a contradiction, as σ(a) 6∈ F . If τ( 4 σ(a)) = √ p √ p 4 4 ± √a, then τ( σ(a)) = a, a contradiction to (∗). Thus τ( σ(a)) = ±i 4 a. • τ(i) = σ(i) = −i, as i 6∈ F . Hence there are 4 possibilities remaining how τ can operate on B0: √ τ p τ √ τ p τ √ √ 4 a 7→ 4 σ(a) 7→ i 4 a 7→ −i 4 σ(a) 7→ i2 4 a = − 4 a √ τ p τ √ τ p τ √ √ 4 a 7→ 4 σ(a) 7→ −i 4 a 7→ i 4 σ(a) 7→ (−i)2 4 a = − 4 a √ τ p τ √ τ p τ √ √ 4 a 7→ − 4 σ(a) 7→ −i 4 a 7→ −i 4 σ(a) 7→ i2 4 a = − 4 a √ τ p τ √ τ p τ √ √ 4 a 7→ − 4 σ(a) 7→ i 4 a 7→ i 4 σ(a) 7→ (−i)2 4 a = − 4 a √ √ In all 4 cases, τ 4( 4 a) 6= 4 a, thus τ has order at least 8. This proves the claim. Now consider the splitting ﬁeld M of M 0 over K. It has degree at most (4n)! over K, and all extensions of σ to M have order at least 8. ¥ Now we proceed to the Proof of Theorem 3.10. Proof of theorem 3.10: Let P0 := {n ≥ 3 | n prime } ∪ {4}, let r(n) := n2 · (n4)! and for all p ∈ P0, let ϕp(x) be a formula expressing that . if ord(x) = p, then there is [z] J³ [x] such that all preimages of x in [z] have order > p, where z is a variable of sort r(n) and x is of sort n. Further let V ϕn := ∀x ϕp(x). p∈P0 p≤n

2 For n ∈ N, let s(n) := n · (n )! and let ψn be a sentence expressing that

if x ∼ y are of order 2 with x 6= y, then there is a [z] projecting on [x] such that for all preimages x0, y0 of x, y respectively in [z] the following holds: At least one out of {x0, y0, x0y0} has order ≥ 4,

43 where x, y are variables of sort n, and x0, y0, z are of sort s(n). Finally let

ΦAS := {ϕn ∧ ψn | n ∈ N}.

First we show part 1) of the theorem. Let S(G) ² ΦAS; we need to show that (AS) holds in G. 1) Assume there is g ∈ G of finite order p 6= 1, 2. By taking an appropriate power of g, we can assume that p ∈ P0. As G is Hausdorff, there is an open normal subgroup N C G such that ord(gN) = p; let n = |G/N|. Then by Lemma 2.17, for all open normal subgroups M C N there exists a z ∈ [M] with πMN (z) = y and ord(z) = ord(y); a contradiction to the validity of ϕn. 2) Let g 6= h ∈ G be elements of order 2. As G is Hausdorff, there is an open normal subgroup N of G such that ord(gN) = ord(hN) = 2 and gN 6= hN. Let n = |G/N|. As S(G) ² ψn and gN 6= hN have order 2, there exists an open normal subgroup M of G, M ≤ N, |G/M| ≤ s(n) such that either gM or hM or ghM have order ≥ 4. As ord(g) = ord(h) = 2, it follows that ghM has order ≥ 4. Thus gh has order ≥ 4, and so by 1) infinite order.

Now we show part 2) of the theorem. Let G = GK be the absolute Galois group of a field K; we show that S(G) ² ΦAS. Because of Theorem 3.11, we can assume that K has characteristic 0. 0 First we show the validity of the ϕn in S(G). Let L /K be a Galois extension of degree n, and let σ ∈ Gal(L0/K) be an element of order p for some p ∈ P0. We have to find a Galois extension M/K of degree ≤ r(n) such that all extensions of σ to M have order > p. 0 First assume that p is a prime ≥ 3. Let L := L (ζp2 ). Then L/K is Galois of degree m ≤ np(p − 1). There are at most p(p − 1) extensions σe of σ to L; for every σe of order p, let Mσe be the Galois extension of degree [Mσe : K] ≤ (pm)! from Lemma 3.13 such that all extensions of σe to Mσe have order > p. Let M be 2 the field generated by all Mσe. Then M/K is normal, and [M : K] ≤ p ·(pm)! ≤ p2 · (np3)! ≤ n2(n4)!, and all extensions of σ to M have order > p. 0 If p = 4, let L := L (ζ8). Then L/K is Galois of degree m ≤ 4n. There are at most 4 extensions σe of σ to L; for every σe of order 4, let Mσe be the Galois extension of degree [Mσe : K] ≤ (4m)! from Lemma 3.15 such that all extensions of σe to Mσe have order > 4. Let M be the field generated by all Mσe. Then M/K is normal, and [M : K] ≤ 4 · (4m)! ≤ 4 · (16n)! ≤ n · (n3)! ≤ n2 · (n4)!, and all extensions of σ to M have order > 4. 0 Finally, we show the validity of the ψn in S(G): Let L /K be a Galois extension 0 of degree n, and let σ1 6= σ2 ∈ Gal(L /K) be elements of order 2. Let i be a primitive 4-th root of unity. 0 If i ∈ L , let Fk for k = 1, 2 be the fixed field of σk and F12 be the fixed field of σ1σ2. Then F1 or F2 or F12 contains i and we can apply Lemma 3.14: There is a Galois extension M ⊇ L0 of degree at most (2n)! such that either all extensions of σ1 or all extensions of σ2 or all extensions of σ1σ2 have degree ≥ 4. 0 0 If i 6∈ L , let L := L (i); then L/K has degree at most 2n. Let τ1, τ2 be arbitrary extensions of σ1, σ2 to L (these have automatically order 2 again). Let Fk for k = 1, 2 be the fixed field of τk and F12 be the fixed field of τ1τ2. Then F1 or F2 or F12 contains i and we can apply Lemma 3.14: There is a Galois extension

Mτ1τ2 ⊇ K of degree at most (4n)! such that all extensions of τ1 or all extensions

44 of τ2 or all extensions of τ1τ2 to Mτ1τ2 have degree ≥ 4. Now let M be the field generated by all Mτ1τ2 , where τ1, τ2 are extensions of order 2 of σ1, σ2 to L. It is a normal extension, and as there are at most 4 such pairs τ1, τ2, it has degree at most 4 · (4n)! ≤ n · (n2)!. Consider arbitrary extensions σf1, σf2 of σ1, σ2 to M respectively. Let τk = σfk . |L Then all extensions of τ1 or all extensions of τ2 or all extensions of τ1τ2 to M have degree ≥ 4. In particular σf1 or σf2 or σf1σf2 have order ≥ 4. This finishes the prove of Theorem 3.10. ¥ Recall that a field K is formally real if and only if −1 is not a sum of squares in K. Denote the class of formally real fields by KFR and let K¬FR be the complement of this class in the class of all fields. KFR is (in the usual language LR of rings) obviously an elementary class. In contrast we have: Proposition 3.16.

1) The class K¬FR is not elementary. 2) The class of complete inverse systems of absolute Galois groups of non- formally real ﬁelds is not elementary. 3) The class of complete inverse systems of torsion free proﬁnite groups is not elementary.

Proof: For a non-formally real field K, let the level s(K) be the minimal natural number n such that −1 can be written as a sum of n squares; if K is formally i real, let s(K) = ∞. Let Ki be a field with s(Ki) = 2 ; such fields exist by a well-knownQ result of Pfister, see for example [Lam]. Then a free ultraproduct K = Ki/U of the Ki has infinite level, hence K is formally real. This proves that the class K¬FR is not closed under ultraproducts, and hence not elementary. Together with Lemma 3.2, this implies that the class of complete inverse systems of absolute Galois groups of non-formally real fields is not elementary: We have ∼ Q S(GK ) = S(GKi )/U, what proves 2).

Finally, as all of the GKi are torsion free, but GK is not, this also shows that the class of complete inverse systems of torsion free proﬁnite groups is not elementary. ¥

Theorem 3.17. There is a set of LIS-sentences ΦFR such that for all ﬁelds K the following holds:

S(GK ) ² ΦFR ⇐⇒ K is formally real.

Proof: Let x be a variable of sort 2 and y be a variable of sort n. Further ∼ let ϕn(x) be the formula stating that [x] = Z/2 and that for every [y] ³ [x] the short exact sequence 0 → N → [y] → [x] → 0 splits. Note that ϕn+1 implies ϕn. Let ψn := ∃xϕn(x) and ΦFR := {ψn | n ∈ N}. Then the absolute Galois group GK of a formally real field K obviously fulfills ΦFR: Write ∼ GK = H o hgi for an element g of order 2. Let M := GK(i); then [M] = Z/2 and gM has order 2. Further for every L ⊇ K(i), there is a complement Kr of K(i) in L (namely the relative real closure of K in L), and thus every sequence 0 → N → [GL] → [gM] → 0 splits. For the other direction, let GK ² ΦFR. Consider a sufficiently saturated ultra- U power K Â K. Then by Theorem 3.8 there is an element a ∈ S(GKU ) Â S(GK )

45 such that a ² ϕn(x) for all n. By Lemma 2.16, the group GKU contains an element of order 2, thus KU is formally real. But then in particular K ⊆ KU is formally real. ¥ Remark: Instead of using the above model theoretic argument in the proof that K is formally real, one could also use purely group theoretic arguments to ﬁnd an element of order 2 in GK .

3.3 Searching for the cyclotomic quotient or “dead ends’ death”

In the entire subsection, let p 6= 2 be a prime. We will in this and in the next section use the facts about Demushkin groups which are stated in Section 4.1 and in Section 4.2. Definition 3.18. Let K be a field of characteristic 6= p containing a primitive p-th root of unity. The maximal cyclotomic p-extension is the extension a K(µp∞ (K ))/K; we call the corresponding quotient of GK , which is also a quotient of GK (p), the maximal cyclotomic p-quotient of GK , respectively of GK (p). Every subextension of the maximal cyclotomic p-extension is called a cyclotomic p-extension; the corresponding quotients of GK , respectively of GK (p), are called cyclotomic p-quotients. Definition 3.19. Let K be a field and let G be a profinite group. An extension L/K is called a G-extension if L/K is a Galois extension with Galois group (isomorphic to) G.

For several questions about axiomatizability (for example, when looking for an axiomatization of Whaples’ theorem, see Theorem 3.31) it seems to be impor- tant whether the absolute Galois group GK of a field K can “see” how many primitive pn-th power roots of unity are contained in K, and which quotient of GK is the maximal cyclotomic p-quotient. A group theoretic characterization of the maximal cyclotomic p-quotient of GK exists for various fields K; but one can easily find examples where no such characterization exists: For example, one can realize the (projective) group Zp as an absolute Galois group of fields containing arbitrarily many p-th power roots of unity, cf. Example 3.30. The cyclotomic extension can thus correspond to the trivial quotient Zp/Zp (if a a µp∞ (K ) = µp∞ (K)) or to the maximal quotient Zp/1 (if µp∞ (K ) 6= µp∞ (K)). More general, the characterization we will in the following use for valued fields and Demushkin fields (cf. Corollary 3.24 and Proposition 3.25) does obviously fail for projective groups. We will finally show that the characterization even fails in the non-projective case (Example 3.30); before, we give two examples of classes of fields in which the maximal cyclotomic p-quotient of GK is defined by group theoretical means. Definition 3.20. 1) For a profinite group G such that the maximal pro-p-quotient G(p) of G is non-trivial, let ½ ¯ ¾ ¯ for all cyclic G/N, |G/N| < pn there exists h (G) := max n ≥ 1¯ p ¯ M CG, M ≤ N s.t. G/M is cyclic of order pn

46 if the maximum exists, and let hp(G) := ∞ otherwise. We call hp(G) the abelian p-height of G. 2) For a ﬁeld K of characteristic 6= p let

up(K) = max {n ∈ N | ζpn ∈ K}

if the maximum exists, and let up(G) := ∞ otherwise. We call up(G) the cyclotomic p-height of K.

The field K contains a primitive p-th root of unity if and only if up(K) ≥ 1. We will in the sequel only consider fields of characteristic 6= p with up(K) ≥ 1; this implies that adjoining all pk-th roots of unity is a p-extension of K. Remark and Definition 3.21. Let G be a profinite group and let K be a field such that GK (p) 6= 1 and up(K) ≥ 1; let char(K) 6= p. 1) We call a Z/p k-extension of K which cannot be extended to a Z/pk+1- extension a (pk)-dead end of K. Similarly, we call a cyclic quotient G/N =∼ Z/p k of G such that there is no M C G, M ≤ N with G/M =∼ Z/p k+1 a (pk)-dead end of G. n 2) hp(G) = n if and only if there is a quotient of G which is a p -dead end, but no quotient of G is a pk-dead end for some k < n. 3) Kummer Theory shows that up(K) ≤ hp(GK ): If ζpn ∈ K is a primitive n k p -th root√ of unity and if L/K is a Z/p√ -extension for some k < n, then L = K( pk a) for some a ∈ K, and L( pk+1 a) is the desired Z/p k+1-extension. k 4) For every k ≥ 1, there is an LIS-formula ϕdead(x) with x a variable of sort pk such that

k k S(G) ² ϕdead(gN) ⇐⇒ G/N is a p -dead end.

k,m 5) For all k, m ≥ 1, there is an LIS-formula ϕlift (x) with x a variable of sort pm such that

k,m ∼ m k+m S(G) ² ϕlift (gN) ⇐⇒ G/N = Z/p and hp(N) = p . If abelian and cyclotomic p-height coincide, then the cyclotomic p-quotients might be deﬁned group theoretically (and uniformly in all such ﬁelds):

Observation 3.22. Assume that K is a class of ﬁelds such that up(K) ≥ 1 for all K ∈ K, and such that hp(GL) = up(L) for all ﬁnite p-extensions L/K of k,m m K ∈ K. Then there is an LIS-formula χ (x) with x a variable of sort p such that for all K ∈ K the following holds:

k,m S(GK ) ² χ (gN) ⇐⇒ up(K) = k and Fix(N) = K(ζpm+k ).

Proof: In an arbitrary field K of characteristic 6= p with 1 ≤ up(K) = k < ∞, m the cyclotomic p-extension L = K(ζpm+k ) of degree p of K is the unique m extension L of degree p of K such that up(L) = m + k. If K is a class of fields such that a primitive p-th root of unity ζp ∈ K for all K ∈ K, and such that hp(GL) = up(L) for all finite p-extensions L/K of K ∈ K, then this is by the last remark in fact a group theoretic statement. ¥ Now we will indicate two natural classes of fields in which the abelian p-height of the absolute Galois group and the cyclotomic p-height of the field coincide.

47 Proposition 3.23 (Cf. Observation 2.24 in [K1]). Let (K, v) be a valued ﬁeld with up(K) ≥ 1, assume that p 6= char(Kv) and that Γ 6= pΓ. Suppose further that up(K) = up(Kv), which holds for example if (K, v) is henselian. Then hp(GK ) = up(K).

Proof: The inequality up(K) ≤ hp(GK ) is clear by Remark 3.21.3). For the other direction,√ let up(K) = k and let a ∈ K be an element with v(a) 6∈ pΓ. We pk k show that K( a)/K is a p -dead√ end. Suppose not. Then L = K( pk a) is contained in a cyclic extension L0/K of degree pk+1. Let w be an extension of v to L0. The extension L/K is totally and tamely ramified. This implies that also L0/K is totally and tamely ramified: 0 Otherwise the inertia field KT of Gal(L /K) would be a complement to L in L0, and Gal(L0/K) would not be cyclic. Thus for the inertia group we have 0 0 0 T (w/v) = Gal(L /K). As Kv = Lw and up(Kv) = k, the group µp∞ (Lw) is isomorphic to Z/p k. Now Theorem 1.48 leads to ∼ k+1 ∼ 0 ∼ k+1 k ∼ k T (w/v) = Z/p = Hom(Γw/Γv, µp∞ (Lw)) = Hom(Z/p , Z/p ) = Z/p , a contradiction. ¥ Proposition 3.24. Let K be a class of valued fields such that all (K, v) ∈ K fulfill the conditions from Proposition 3.23. Further let up(L) = up(Lw) for all finite p-extensions (L, w)/(K, v), which holds for example if all fields in K k,m are henselian. Then for all k, m ≥ 1 there is an LIS-formula χ (x) with x a variable of sort pm such that for all K ∈ K the following holds:

k,m S(GK ) ² χ (gN) ⇐⇒ up(K) = k and Fix(N) = K(ζpm+k ).

Proof: This follows directly from Observation 3.22 and Proposition 3.23. ¥ Another example for absolute Galois groups that “see” the cyclotomic p-extensions are the absolute Galois groups of so-called Demushkin fields (cf. Definition 4.8:(p-)Demushkin fields are fields K of characteristic 0 with up(K) ≥ 1 such that GK (p) is a Demushkin group). All these fields satisfy hp(GK ) = up(K), see Corollary 4.10. As open subgroups Demushkin groups are again Demushkin groups (see for example [Se], Chapitre I.4.5), this equality also holds for finite extensions of K. So one gets: Proposition 3.25. Let K be the class of (p-)Demushkin fields. Then there is k,m m an LIS-formula χ (x) with x a variable of sort p such that for all K ∈ K holds:

k,m S(GK ) ² χ (gN) ⇐⇒ up(K) = k and Fix(N) = K(ζpm+k ).

Note that χk,m is the same formula as in Observation 3.22 and Proposition 3.24. Unfortunately, Example 3.30 will show that this method does not allow to characterize the cyclotomic pm-quotient in every non-projective absolute Ga- lois group. To give the example, we need several valuation theoretic lemmata in which fields with a prescribed Galois group are constructed. The lemmata are more or less well-known; but lacking suitable references for the first two of them, we give sketches of proofs. In particular, there are no references paying attention to the question which roots of unity are contained in the current field, and this question is crucial for our needs.

48 Lemma 3.26. Let K be a ﬁeld with char(K) = 0 and up(K) ≥ 1, let K have transcendence degree trQ(K) = α over Q, and let β ≥ α. Then there is L ⊇ K ∼ with trQ(L) = β, such that GL = GK and up(L) = up(K).

Sketch of Proof (See Picture 3.26): Let t¯ be a transcendence basis of K over Q, and let s¯ be a tuple of algebraically independent elements over Q(t¯) such that trQ(Q(t,¯ s¯)) = β, where the lengths of t,¯ s¯ can be arbitrary cardinals. Let v be the trivial valuation on Q(t¯); using repeatedly Lemma 1.44, one finds an extension of v to Q(t,¯ s¯) with residue field Q(t¯). Fix some extension of this valuation to Q(t,¯ s¯)a; by abuse of notation, we denote all restrictions of this valuation to intermediate fields between Q(t,¯ s¯) and Q(t,¯ s¯)a again by v. Let Q be a maximal purely ramified extension of a henselization of (Q(t,¯ s¯), v). Then the residue field of Q is Q(t¯) and the value group is p-divisible; thus the extension Qa/Q is purely inert. Theorem 1.47 then provides an isomorphism Φ: GQ → GQ(t¯). Let −1 H := Φ (GK ) and L = Fix(H); it follows that L has residue field K. As L is again henselian valued, it follows from Hensel’s Lemma that up(L) = up(K).

Residue Valued ﬁeld ﬁeld

Ka (Q(t,¯ s¯)a, v)

GK =∼ GK

purely K (L, v) := (Fix(H), v) inert

Q(t¯) (Q, v)

Q(t¯) (Q(t,¯ s¯), v)

Picture 3.26

Lemma 3.27. Let K1,K2 be ﬁelds of characteristic 0 with up(K1), up(K2) ≥ 1.

Then there exists a twofold valued ﬁeld (L, v1, v2) with Lv1 ,Lv2 ﬁelds of characteristic 0 such that G =∼ G and u (K ) = u (L ) for i = 1, 2. Ki Lvi p i p vi

0 0 Sketch of proof (See Picture 3.27): First use Lemma 3.26 to ﬁnd ﬁelds K1,K2

49 0 0 with transcendence degree tr (K ) = tr (K ) ≥ 2 such that G 0 =∼ G and Q 1 Q 2 Ki Ki 0 0 0 up(Ki) = up(Ki) for i = 1, 2. Thus we can assume that K1,K2 are algebraic over Q(t,¯ s1) and Q(t,¯ s2) respectively, where t¯ is a tuple of algebraically independent elements of adequate length, and s1, s2 are algebraically independent over Q(t¯). Let Q := Q(t,¯ s1, s2), let v1 be the s2-adic and v2 be the s1-adic valuation on

Q (thus Γv1 = Z, Γv2 = Z, v1(s2) = 1, v1(s1) = 0, v2(s1) = 1, v2(s2) = 0 and v1, v2 are trivial on Q(t¯)). Then v1, v2 are independent, and for the residue ∼ ¯ fields we have Qvi = Q(t, si). Choose two arbitrary extensions of v1, v2 to the algebraic closure Qa of Q; by abuse of notation, we denote all restrictions of a these valuations to intermediate fields between Q and Q again by v1, v2. All these extensions are independent again. Now let L be a maximal subfield of Qa 0 0 such that the residue fields L1,L2 belonging to v1, v2 embed into K1,K2. We 0 claim that Li = Ki: 0 Assume not. Then there is without loss of generality a¯ ∈ K1 \ L1. Let a be a 0a a preimage of a¯ under the place ϕv1 : Q → K1 belonging to v1. Use the a Approximation Theorem to find l ∈ Q such that l − a ∈ Mv1 and l − 1 ∈ Mv2 . 0 Then L(l)v1 = L1(¯a) embeds into K1 and L(l)v2 = L2, thus L was not maximal.

Residue Valued Residue field field field

a (Q , v1, v2)

0 0 K1 K2

L1(¯a) (L(l), v1 , v2 ) L2 |L(l) |L(l)

L1 (L, v1 , v2 ) L2 |L |L

Q(t,¯ s1) (Q(t,¯ s1, s2), v1, v2) Q(t,¯ s2) Q

Picture 3.27

Lemma 3.28. Let (L, v1, v2) be a twofold valued field with char(Lv1 ) = char(Lv2 ) = 0. Then there is an extension (F, w1, w2)/(L, v1, v2) of twofold valued fields such that G =∼ G ∗ G and u (F ) = min{u (L ), u (L )}. F Lv1 Lv2 p p v1 p v2 Proof: This follows from the proof of Proposition 3.7 in [K2]: Given a twofold valued field (L, v1, v2) with decomposition subgroups D1,D2, there is an exten-

50 ∼ sion (F, w1, w2)/(L, v1, v2) such that GF = D1 ∗D2; the proof of the proposition shows that F = F1 ∩ F2 for henselian valued ﬁelds (Fi, wei) with (Fi)wei = Lvi .

It follows from Hensel’s Lemma that up(Fi) = up(Lvi ), and thus up(F ) = min{up(Lv1 ), up(Lv1 )}. ¥ As a corollary, we get the following well-known fact: Corollary 3.29. The class of absolute Galois groups is closed under free products.

Proof: Let G1,G2 be absolute Galois groups. Realize G1,G2 in characteristic 0 using Theorem 3.11, and use Lemma 3.27 to ﬁnd a twofold valued ﬁeld (L, v1, v2) with G = G and G = G . The result follows with Lemma 3.28. ¥ Lv1 1 Lv2 2

Example 3.30. Let i ≥ 1. Then there are ﬁelds F1,...,Fi and a proﬁnite group G such that ∼ 1) GFl = G for all 1 ≤ l ≤ i. 2) G is not projective. 3) up(Fl) = l for 1 ≤ l ≤ i.

Proof: Fix i > 1. Then there is n ≥ 2 and an algebraic extension K1 of ∼ Qp(ζp) such that GK1 (p) = Dn,i, where Dn,i is a pro-p Demushkin group with i n generators and q(Dn,i) = p , cf. Section 4.2. It follows that up(K1) = i. Let

GK1 =: H. For every l ∈ {1, . . . , i} let now K2/Q(ζpl ) be a maximal algebraic extension ∼ such that ζpl+1 6∈ K2; such an extension exists, and we have GK2 = Zp and up(K2) = l, cf. Theorems 2 and 5 in [Q]. Using Lemma 3.27, one gets a twofold valued field (L, v , v ) such that G ∼ H, G ∼ Z and u (L ) = i, 1 2 Lv1 = Lv2 = p p v1 up(Lv2 ) = l. Using Lemma 3.28, we find an extension (Fl, w1, w2)/(L, v1, v2) of ∼ twofold valued fields such that GFl = H ∗ Zp =: G, and up(Fl) = min{l, i} = l. Then 1) and 3) are fulfilled, and G is not projective: As subgroups of projective groups are projective again, with G also H would have to be projective. One can easily show that then in particular the maximal pro-p quotient Dn,i of G would have to be projective, but this is not the case. ¥

Considering the Fl for 1 ≤ l ≤ i, we have found fields with isomorphic absolute Galois groups, and thus with same abelian height, but with different cyclotomic heights; thus our characterization of the cyclotomic extension from Lemma 3.22 does not work for all fields with non-projective Galois group. This last example is a serious obstacle for the search for an axiomatization of the property of absolute Galois groups stated in the next theorem. Recall that p =6 2. Theorem 3.31 (Whaples). If the field K has a cyclic extension of degree p, then it has a Zp-extension.

Let (W) be the property (of a profinite group) “if G has a continuous quotient isomorphic to Z/p , then it has a continuous quotient isomorphic to Zp”. The class of complete inverse systems of arbitrary profinite groups fulfilling (W) is not axiomatizable: For example, the complete inverse systems of the groups Q n+1 Q n+1 G1 = n∈N Z/p and G2 = n∈N Z/p × Zp are elementarily equivalent (cf. Theorem 5.2) and (W) holds in G2, but not in G1.

51 The proof of Whaples’ theorem roughly goes as follows (the case that the ﬁeld does not contain a primitive p-th root of unity is slightly more complicate): If the ﬁeld K does not contain all pk-th roots of unity, then the maximal cyclotomic p-extension is the desired Zp-extension. Otherwise one uses Kummer Theory to extract step by step all pk-th roots of unity of an element a which is not a p-th power in K. There might still exist a property (W’) true in all complete inverse systems S of absolute Galois groups, and which implies (W) for the groups G(S). Whether or not such (W’) exists is an open question. But the last example shows that we cannot easily imitate the case distinction from the proof of Whaples’ theorem in the desired axiomatization. In fact, this weakness of the expressive power of LIS might provide a possibility to prove that the class AG is not axiomatizable, compare with the explanations on page 62.

3.4 The Elementary Type Conjecture

The “Elementary Type Conjecture” (ETC from now on) guesses which finitely generated pro-p-groups occur as absolute pro-p Galois groups. In the first subsection, we do the necessary preliminary work to be able to formulate the ETC. In the second subsection, we set up the technical framework required in the third subsection, in which we prove the axiomatizability of the class of groups described in the ETC. Notations: In the whole section, we fix a prime p 6= 2. We will write ∗ instead of ∗p to denote the free pro-p-product. Further let N≥1 := N \{0} and N≥1 := N≥1 ∪ {∞}. We consider N≥1 as ordered with ∞ as maximal element. Recall that 1 + pZp is a multiplicative subgroup of the ring Zp.

3.4.1 ETC – Statement and motivation

We need some preparatory work to be able to formulate the ETC. For the basics about Demushkin groups, we refer to the Sections 4.1 and 4.2. Compare also with the presentation of the subject in [Ef3], Section 3. Deﬁnition 3.32. A cyclotomic pair (G, χ) consists of a ﬁnitely generated pro- p-group G and a continuous homomorphism χ : G → 1 + pZp. We call χ the character of (G, χ).

A homomorphism f :(G, χ) → (G0, χ0) of cyclotomic pairs is a continuous homomorphism f : G → G0 such that χ = χ0 ◦ f. As usual in algebra, we will often not distinguish between isomorphic cyclotomic pairs, or between a pair and its isomorphism type. The next example explains the name “cyclotomic pair”: Example and Definition 3.33. Let K be a field of characteristic 0 containing a primitive p-th root of unity. We define the cyclotomic pair (GK (p), χK ), the Galois pair of K, as follows: a Let µp∞ = µp∞ (K ). Let r : GK (p) → Aut(µp∞ /µp) be the restriction map. ∼ There is a natural isomorphism ϕ : 1 + pZp = Aut(µp∞ /µp), where ϕ(α) for

52 k α ∈ 1 + pZp is deﬁned as follows: If ζ = ζpk is a p -th root of unity, if πi = i α Zp → Z/p is the natural projection and if πi(α) = αi, then ϕ(α)(ζ) = ζ k . −1 Finally χK = ϕ ◦ r.

For i ≥ 1 and α ∈ 1 + pZp, there is exactly one continuous homomorphism χα : Zp → 1 + pZp such that χα(1) = α. Further if im(χα) = im(χβ), then the cyclotomic pairs (Zp, χα) and (Zp, χβ) are isomorphic. Note that as 1 + pZp ⊆ × ∼ Zp = Aut(Zp), the character χ of a cyclotomic pair (G, χ) defines an operation of G on Zp (given by g.z = χ(g) · z). Now we define some special types of cyclotomic pairs: Definition 3.34 (Basic cyclotomic pairs). 1) Let (1, 1) be the trivial cyclotomic pair. 2) For a Demushkin group D, let χD be the homomorphism from Theorem 4.7 and (D, χD) be the corresponding cyclotomic pair. Let D be the class of pairs 2 (D, χD), where D is a p-adic Demushkin group . 3) For every i ≥ 1, let (Zp, χi) be the (up to isomorphism) unique cyclotomic i pair such that im(χi) = 1 + p Zp. Denote by (Zp, χ∞) the pair with the trivial character. Let Z be the class of cyclotomic pairs (Zp, χi) for some i ∈ N≥1. Remark 3.35. p-adic Demushkin groups fulfill certain extra conditions connecting the invariants n(D) and q(D) defined in Section 4.2. In particular, if D is p-adic Demushkin of rank n = n(D) with q(D) = pi, then one always has p−1 n(D) − 2 ≥ q p . As the invariants n(D) and q(D) determine D up to isomorphism (cf. Theorem 4.3), there are only finitely many p-adic Demushkin groups of fixed rank n. Definition 3.36 (Products of cyclotomic pairs). 1) If (G1, χ1) and (G2, χ2) are cyclotomic pairs, then the free cyclotomic product of (G1, χ1) and (G2, χ2) is the cyclotomic pair (G1 ∗ G2, χ1 ∗ χ2), where χ1 ∗ χ2 : G1 ∗ G2 → 1 + pZp is the homomorphism given by the universal property of the free pro-p product. 2) If (G, χ) is a cyclotomic pair, let Zp oχ G be the semidirect product with respect to the operation of G on Zp defined by χ. Let π : Zp oχ G → G be the natural projection. Then Zp oχ G together with the homomorphism χ ◦ π (which we denote by abuse of notation again by χ) is called the semidirect cyclotomic product (Zp oχ G, χ) of (G, χ) and Zp. Remark 3.37. • We will use an a bit unconventional notation for the multiplication in semidirect cyclotomic products: We write the group operation in the commutative component Zp additively, but both in G and Zp oχ G, we use multiplicative 0 0 0 0 notation. For example: If (z, g), (z , g ) ∈ Zp oχ G, we write (z, g)(z , g ) = (χ(g0)z + z0, gg0). ∼ 2 2 • Note that (Zp oχi (Zp oχi Zp, χi)) = (Zp oχi Zp, χi), where (Zp oχi Zp, χi) is 2 the semidirect product induced by the componentwise operation of Zp on Zp via χi. Definition 3.38. 2A p-adic Demushkin group is a Demushkin group that is realized as an absolute pro-p- Galois group of a p-adic field, see Definition 4.12.

53 1) Let P be the smallest class of cyclotomic pairs containing the trivial cyclotomic pair (1, 1) and the sets D and Z, that is closed under free cyclotomic products and semidirect cyclotomic products. 2) Let Pn be the set of cyclotomic pairs (G, χ) in P such that G has rank n. 3) Let G := {G | there is χ such that (G, χ) ∈ P} and Gn := {G ∈ G | rank(G) = n}. 4) Finally let S := {S(G) | G ∈ G} and Sn := {S(G) | G ∈ Gn}. Example 3.39. 1) P1 = Z.

2) P2 = {(Zp oχi Zp, χi) | i ∈ N≥1} ∪ {(Zp ∗ Zp, χi ∗ χj) | i, j ∈ N≥1}. 3) P3 = A1 ∪ A2 ∪ A3 ∪ A4, where

• A1 = {(Zp oχi (Zp oχi Zp), χi) | i ∈ N≥1},

• A2 = {(Zp oχi∗χj (Zp ∗ Zp, χi ∗ χj) | i, j ∈ N≥1}, • A3 = {(Zp ∗ Zp ∗ Zp, χi ∗ χj ∗ χk) | i, j, k ∈ N≥1},

• A4 = {(Zp ∗ (Zp oχi Zp), χi ∗ χj) | i, j ∈ N≥1}. 4) If n < p + 1, then Pn does not contain a Demushkin group, see Remark 3.35. Pp+1 contains exactly one Demushkin group, namely the group D with invariants n(D) = p + 1 and q(D) = p (which is realized over Qp(ζp)). Conjecture 3.40 (Elementary Type Conjecture, ETC). A cyclotomic pair (G, χ) is isomorphic to a Galois pair (GK (p), χK ) if and only if (G, χ) ∈ P. ∼ In particular, a ﬁnitely generated pro-p-group G has a realization G = GK (p) as an absolute pro-p Galois group if and only if G ∈ G.

The ETC is proven for groups of rank ≤ 4 and p 6= 3 (see [K4], Theorems 1.1 – 1.3, or [Ef2], Section 5). For rank 4 and p = 3, the validity depends on the question whether there exists a non-p-adic Demushkin group of rank 4 that is an absolute pro-p Galois group of a ﬁeld of characteristic 0, cf. Conjecture 4.13. Further it is known that all pairs in P are realized as Galois pairs (Remark 3.4 in [Ef]). The rough idea to prove this is as follows: • For p-adic Demushkin groups, one just has to check that the cyclotomic characters are the right ones, cf. Corollary 4.11. To realize the pairs from Z, one can for example use the construction from Corollary 3.30. • The realization of the free pro-p-products can be deduced from Corollary 3.29, as the maximal pro-p-quotient of G1 ∗G2 is the free pro-p-product of the maximal pro-p-quotients of G1 and G2. • If G is realized over K with char(K) = 0 with corresponding homomorphism χ, then Zp oχ G is realized over the henselization of K((t)) with respect to the t-adic valuation.

3.4.2 The technical framework for the ETC

The axiomatization of the class G in the next subsection is based on a description of the iterated Frattini quotients of the groups in G; therefore, we will need more information about the Frattini subgroups and quotients of these groups. The subsection is of a quite technical nature; not all proofs are performed in detail. We start with a lemma describing the Frattini series of cyclotomic semidirect products. The proof is an easy but lengthy calculation which we will only sketch.

54 Lemma 3.41. Let (G, χ) be a cyclotomic pair. Consider (z, g) ∈ Zp oχ G. Then 1) (z, g)−1 = (−χ(g−1)z, g−1). p Pp−1 i p p 2) (z, g) = ( i=0 χ(g) z, g ) ∈ {(pz, g ) | z ∈ Zp, g ∈ G}. p p 3) (Zp oχ G) = pZp oχ G . |Gp 4) [Zp oχ G, Zp oχ G] ⊆ pZp oχ [G, G]. |[G,G] 5) Φ(Zp oχ G) = pZp oχ Φ(G). |Φ(G) l l l 6) Similarly for l ≥ 1: Φ (Zp oχ G) = p Zp oχ Φ (G). |Φl(G)

Sketch of proof: 1) (−χ(g−1)z, g−1)(z, g) = (−χ(g)χ(g−1)z + z, g−1g) = (0, 1). p Pp−1 i p 2) If χ(g) = 1 + pα for some α ∈ Zp, then (z, g) = ( i=0 χ(g) z, g ) = Pp−1 i p p ( i=0 (1 + pα) z, g ) = (pz + pβz, g ) for an adequate β ∈ Zp. p 3) First one easily sees that pZp oχ G is a subgroup. This together with |Gp p p p 2) shows (Zp oχ G) ⊆ pZp oχ G . For (pz, g) ∈ pZp oχ G we have |Gp |Gp p (pz, g) = (0, g)(pz, 1) ∈ (Zp oχ G) . 4) [(z, g), (z0, g0)] = (x, [g, g0]), where

0 0 x = (χ(g ) − 1)z + (1 − χ(g))z ∈ pZp.

Further pZpoχ [G, G] is easily seen to be a subgroup, thus [ZpoχG, Zpoχ |[G,G] G] ⊆ pZp oχ [G, G]. |[G,G] 5) Follows easily with 3) and 4), and 6) follows then inductively. ¥ For the next corollary, recall that F i(G) := G/Φi(G) is the i-th Frattini quotient of G. Corollary 3.42. Let (G, χ) be a cyclotomic pair and let l ≥ 1. Then

l l l F (Zp oχ G) = Zp/p Zp oχl F (G),

l where oχl is the semidirect product deﬁned by the induced action of F (G) on l l l Zp/p Zp given by Lemma 1.2: If g¯ = g · Φ (G) ∈ F (G) and z ∈ Zp, then l l i g.¯ (z + p Zp) = χ(g)z + p Zp. In particular, if im(χ) = 1 + p Zp and if l < i, then oχl is a direct product.

Now we will give an alternative description of the pairs in P which allows us to make some useful technical deﬁnitions, and to prove properties of groups in G via an inductive principle. Let

i PD := {(D, i) | D p-adic Demushkin, q(D) = p }

PZ := {(Zp, i) | i ∈ N≥1}

P := PD ∪ PZ .

Note that the second component of a pair (D, i) ∈ PD is determined by the ﬁrst component.

55 There is a natural bijection between PD and D and PZ and Z, thus elements of P correspond to nontrivial basic cyclotomic pairs. Because of Remark 3.35, the set PD contains for every n ∈ N only finitely many pairs (D, i) with D a group of rank n. Now we consider labeled rooted finite trees T in which every vertex has at most 2 children. The leaves of T (that is, the vertices of T without children) are labeled with pairs pi from P . We call such a structure a tree structure in the following and denote it by (T, p1, . . . , pk) = (T, p). This notation is not exact; correctly, one would have to consider rooted trees T together with a multiset S of pairs from P and a bijection from S to the leaves of T . We will keep the simplified notation from above to avoid a notational complexity that is disproportionate to the simplicity of the concept, and imagine the leaves of T to be ordered and pi attached to the i-th leaf. Example 3.43. Let D be a p-adic Demushkin group with q(D) = p3. The following picture illustrates the idea of a tree structure:

(D, 3) (Zp, 5) (Zp, ∞)

Denote the tree structure from the picture by V.

Every tree structure (T, p) with more than one vertex originates (depending on whether his root has 1 or 2 children)

• either from two tree structures (T1, p1), (T2, p2), whose roots are the two children of the root of (T, p); we denote (T, p) by (T1, p1) ∗ (T2, p2) in this case. • or from a tree structure (T 0, p0) whose root is the only child of (T, p); we 0 0 denote (T, p) by Zp o (T , p ) in this case. Example 3.44. (Example 3.43 continued) Let V be the tree structure from the last example. Let

• V1 be the tree structure with one vertex labeled with (D, 3), • V2 be the tree structure with one vertex labeled with (Zp, 5), • V3 be the tree structure with one vertex labeled with (Zp, ∞).

Then V = (Zp o V1) ∗ (V2 ∗ V3).

Now we can define or prove properties for arbitrary tree structures by induction on the height of the trees. The induction base will be denoted by [base case] in the following; the two induction steps by [∗-step] and [o-step]. As a first application of this induction principle, we assign to every tree structure (T, p) a cyclotomic pair (G(T, p), χ(T, p)) ∈ P: [base case] To a tree structure with one vertex we assign the cyclotomic pair defined by its label.

56 [∗-step] If (T, p) = (T1, p1) ∗ (T2, p2) and the cyclotomic pairs assigned to (T1, p1), (T2, p2) are (G1, χ1), (G2, χ2), then we assign (G1 ∗G2, χ1 ∗χ2) to (T, p). 0 0 0 0 [o-step] (T, p) = Zp o (T , p ) and the cyclotomic pair assigned to (T , p ) is (G, χ), then we assign (Zp oχ G, χ) to (T, p). Remark and Deﬁnition 3.45. 1) We will in the following call (G(T, p), χ(T, p)) ∈ P the cyclotomic pair attached to (T, p). 2) One can easily show that isomorphic tree structures lead to isomorphic cyclotomic pairs. 3) Every nontrivial cyclotomic pair from P can be represented by at least one3 isomorphism class of tree structures. Example 3.46. (Example 3.43 continued) We use the same deﬁnitions as in the preceding two examples. The cyclotomic pair attached to V is

(Zp oχD (D, χD) ∗ ((Zp, χ5) ∗ (Zp, χ∞)).

Now we deﬁne two tree structures (T, p) and (T 0, q) to be macroequivalent if there is an isomorphism T =∼ T 0 of rooted trees such that if the leaf labeled with pi is mapped on the leaf labeled with qj

• and pi ∈ PD, then pi = qj; whereas if • pi ∈ PZ , then only the ﬁrst components of the pairs pi, qj have to be equal. To be macroequivalent is obviously an equivalence relation. Example 3.47. (Example 3.43 continued) Let V be as in the preceding examples and let V0 be the tree structure

(D, 3) (Zp, 7) (Zp, 1001)

Then V and V0 are macroequivalent. If V00 is the tree structure

0 (D , 2) (Zp, 7) (Zp, 1001)

3At least theoretically, it could be possible that there exist two essentially diﬀerent ways to represent a cyclotomic pair as a tree structure. As we don’t need the uniqueness of the representation, we don’t rack our brains over this question.

57 where D0 is a p-adic Demushkin group with q(D) = p2, then V00 is neither macroequivalent to V nor to V0.

The idea of this equivalence relation is the following: We want to identify cyclotomic pairs that arise in the same way from the same basic groups, if at most the indices i in the pairs (Zp, χi) corresponding to the labels in PZ differ from each other. Denote by Ma(T, p) the equivalence class of (T, p) modulo this equivalence relation; we call this the macrostructure of (T, p). We can recover (T, p) from s Ma(T, p) by specifying the tuple (i1, . . . is) ∈ (N≥1) , where s is the number of leaves of T labeled with a pair from PZ , and where ir is defined as follows: If the r-th leaf of T which is labeled with a pair from PZ is labeled with pl, then ir is 4 the second component of pl. We call the tuple (i1, . . . is) the microstructure of (T, p). Again, this notation is not exact; to be correct, one would have to work with the multiset S = {i1, . . . is} and a map attaching the labels to the leaves. s We call s the length of the microstructure. Every tuple (i1, . . . is) ∈ (N≥1) defines a unique tree structure belonging to Ma(T, p) and (i1, . . . is), and thus a unique cyclotomic pair attached to Ma(T, p) and (i1, . . . is). Example 3.48. (Example 3.43 continued) The microstructure of V from above is (5, ∞).

Lemma 3.49. Let (T, p1, . . . , pk) be a tree structure, let pl = (Gl, il) and let d be the number of vertices of T that have exactly one child. Then

Xk rank(G(T, p)) = rank(Gl) + d. l=1

Proof: [base case] Clear. [∗-step] The rank of a free pro-p-product of pro-p-groups is the sum of the ranks of the factors, see Proposition 9.1.15 in [RZ]. [o-step] It follows from Corollary 3.42 that the rank of a semidirect cyclotomic product Zp o (G, χ) is equal to 1 + rank(G). ¥ Corollary 3.50. 1) The rank of G(T, p) only depends on the equivalence class Ma(T, p). 2) For a ﬁxed natural number n, there are only ﬁnitely many macrostructures such that the corresponding group has rank n.

Proof: 1) follows directly from the last Lemma. For 2), recall that for any given n, there are only ﬁnitely many p-adic Demushkin groups of rank n, cf. Remark 3.35. ¥ The following lemma is central in the axiomatization of P in the next subsection. The main idea behind the lemma (and the next corollary) is that the l-th Frattini quotient cannot distinguish between groups which originate from the

4 Thus in the microstructure, we store the information about the characters χi of the pairs (Zpχi) corresponding to the labels in PZ ; this is exactly the information we loose when passing to the macrostructure.

58 same macrostructures, if the diﬀerences in the microstructures only occur in a “level” larger then l.

Lemma 3.51. Let Ma(T, p) be an arbitrary macrostructure, let ga = (a, g1, . . . , gs) and gb = (b, g1, . . . , gs) be two microstructures for Ma(T, p) and let (Ga, χa) and (Gb, χb) be the corresponding cyclotomic pairs. Further let l < min{a, b}. Then there are free presentations Ga = F/Ra and Gb = F/Rb such that l l 1) Φ (F )Ra = Φ (F )Rb. l 2) If π is the restriction of the natural projection Zp → Zp/p Zp to 1 + pZp and ϕa, ϕb are the lifts of χa, χb to F , then π ◦ ϕa = π ◦ ϕb.

Proof: If G = F/R, we will identify F l(G) and F/Φl(F )R via the canonical isomorphism from Lemma 1.7 to simplify notation; the reader might easily con- vince himself that the above isomorphism is compatible with all maps considered below. We prove 1) and 2) simultaneously by an induction over the macrostructure of (T, p). [base case] We can assume that the only vertex of T is labeled with a pair from PZ (as otherwise the microstructure is empty and the statement is trivial). Thus we can assume that Ga = Gb = Zp = F and Ra = Rb = 1, hence statement 1) is trivial. To prove statement 2), consider the corresponding characters χa, χb. As l < min{a, b}, both π ◦ χa and π ◦ χb are trivial, what proves statement 2).

[∗-step] Let Ma(T1, p1), Ma(T2, p2) be two macrostructures, let ha = (a, g1, . . . , gs) and hb = (b, g1, . . . , gs) be two microstructures for Ma(T1, p1) and let h = (h1, . . . , hr) be a microstructure for Ma(T2, p2). Let (Ha, χa), (Hb, χb) be the cyclotomic pairs attached to Ma(T1, p1) and ha respectively hb, and let (H, χ) be the cyclotomic pair attached to Ma(T2, p2) and h. Let Ha = F1/Ra,Hb = F1/Rb ∼ be free presentations as in the conclusion of the lemma and let H = F2/R be a free presentation of H. We are seeking free presentations of Ha ∗ H and Hb ∗ H such that statements 1) and 2) are fulﬁlled. ∼ ∼ Consider the free presentations Ha ∗ H = F1 ∗ F2/h|Ra,R|i and Hb ∗ H = l l l F1 ∗ F2/h|Rb,R|i, cf. Lemma 1.10. We have Φ (F1 ∗ F2) ⊇ h|Φ (F1), Φ (F2)|i; l l by the induction hypotheses further Φ (F1)Ra = Φ (F1)Rb, thus we get

l l Φ (F1 ∗ F2)h|Ra,R|i = Φ (F1 ∗ F2)h|Ra,R,Rb|i l = Φ (F1 ∗ F2)h|Rb,R|i.

This proves statement 1). To prove statement 2), let ϕa, ϕb, ϕ be the lifts of χa, χb, χ on F1 respectively F2. Consider the commutative diagram

59 F1 ∗ F2

(π ◦ ϕa) ∗ (π ∗ ϕ) = (π ◦ ϕb) ∗ (π ◦ ϕ)

ϕb ∗ ϕ ϕa ∗ ϕ

ϕa π l × F1 1 + pZp (Zp/p Zp)

ϕb

The induction hypotheses implies (τ ◦ ϕa) ∗ (τ ◦ ϕ) = (τ ◦ ϕb) ∗ (τ ◦ ϕ); it follows that τ ◦ (ϕa ∗ ϕ) = τ ◦ (ϕb ∗ ϕ), what proves statement 2).

[o-step] Let Ma(T, p) be a macrostructure and let ga = (a, g1 . . . , gs) and gb = (b, g1, . . . , gs) be two microstructures for Ma(T, p), let (Ga, χa) and (Gb, χb) be the corresponding cyclotomic pairs. Then ga and gb are microstructures for

Ha = Zp oχa Ga and Hb = Zp oχb Gb as well. Let Ga = F/Ra and Gb = F/Rb be free presentations as in the conclusion of the lemma. We are seeking free 0 0 0 0 presentations Ha = F /Ra and Hb = F /Rb such that statements 1) and 2) are fulﬁlled. 0 0 0 We choose F = Zp ∗ F ; let Ra be the kernel of the natural surjection F ³ Ha, 0 which is the identity on Zp and maps F to F/Ra, deﬁne Rb accordingly. As usual, we denote the characters on Ha and Hb again by χa and χb; let ϕa, ϕb be 0 the lifts of χa, χb to F . Now statement 2) follows directly from the induction hypotheses. To prove 1), use Corollary 3.42 and consider the natural surjections of Ha,Hb on the Frattini quotients

l l l H ³ F (H ) = Z /p Z o l F/Φ (F )R and a a p p χa a l l l Hb ³ F (Hb) = Zp/p Zp o l F/Φ (F )Rb; χb

l l By induction hypotheses 1), we have Φ (F )Ra = Φ (F )Rb; by induction hy- l potheses 2), the operations induced by χa, χb on Zp/p Zp are the same. Thus l l l 0 0 l 0 0 F (Ha) = F (Hb). But Φ (F )Ra and Φ (F )Rb are the kernels of the surjec- 0 l 0 l l tions F ³ Ha ³ F (Ha) and F ³ Hb ³ F (Hb) = F (Ha) respectively, l 0 0 l 0 0 hence Φ (F )Ra = Φ (F )Rb. This proves 1) and ﬁnishes the proof. ¥

Corollary 3.52. Let Ma(T, p) be an arbitrary macrostructure, let ga = (a, g1, . . . , gs) and gb = (b, g1, . . . , gs) be two microstructures for Ma(T, p) and let (Ga, χa) and (Gb, χb) be the corresponding cyclotomic pairs. Further let l < min{a, b}. Then with the same identiﬁcation as above

l l F (Ga) = F (Gb).

l l l l l Proof: We have F (Ga) = F (F/Ra) = F/Φ (F )Ra = F/Φ (F )Rb = F (Gb). ¥

60 3.4.3 Axiomatization of the Sn

In this subsection, we will axiomatize the classes Sn for all n ∈ N. Obviously the class S is not axiomatizable, as a free ultraproduct of complete inverse systems of pro-p-groups of increasing rank is the inverse system of a group of =n infinite rank. Recall that we already know that the class AGp of absolute pro- p Galois groups of rank n is axiomatizable, see Corollary 3.7; thus assuming the ETC, we already know that Sn is axiomatizable. But we want to find an axiomatization independent of the validity of the ETC; moreover, we would like to find a (potentially) effective axiomatization.

Theorem 3.53. There is a set Φ of LIS-sentences such that for every pro-p- group G the following holds:

S(G) ² Φ ⇐⇒ G ∈ Gn.

Thus the class Sn is axiomatizable.

Proof: The idea is to describe the iterated Frattini quotients of the groups in Gn. The l-th Frattini quotient of a finitely generated pro-p-group is a finite group of bounded cardinality, cf. Remark 1.5; thus for every l, there are only finitely many possibilities for its isomorphism type. Together with Lemma 2.10, this shows that for every l ∈ N the statement

“[Φl(G)] =∼ F l(H) for some group H ∈ Gn”

5 is an elementary statement about S(G). Let ϕl be the LIS-sentence expressing this, and let Φ = {ϕl | l ∈ N}. Then obviously all complete inverse systems of groups from Gn are models of Φ. What we have to prove is that whenever S(G) ² Φ, then G already lies in Gn. For l ∈ N, let (Hl, χl) ∈ Pn and let G be a group of rank n such that l ∼ l l l F (G) = F (Hl) for all l ∈ N. Let (T , p ) be a tree structure that deﬁnes the cyclotomic pair (Hl, χl). As there are only ﬁnitely many macrostructures of rank n, we can assume that all tree structures (T l, pl) have the same macrostructure l l l l Ma(T , p ) = Ma(T, p); let ¯l = (jl1 , . . . , jls ) be the microstructure of (T , p ). For every l and every r ∈ {l1, . . . , ls} substitute jr by the maximal element kr ∈ N≥1 such that

l ∼ l F (Ma(T, p), (jl1 , . . . , jr, . . . , jls )) = F (Ma(T, p), (jl1 , . . . , kr, . . . , jls )).

Note that this maximum exists by Corollary 3.52: If for every n ∈ N≥1 we l ∼ let Gn be the group attached to (Ma(T, p), (jl1 , . . . , n, . . . , jls )) and if F (Gn) = l l ∼ l F (Gm) for arbitrarily large n, m ∈ N, then also F (Gn) = F (G∞) for all n ≥ 1. ¯ l ∼ l ¯ For the tuple kl = (kl1 , . . . , kls ) we have F (Ma(T, p), ¯l) = F (Ma(T, p), kl). ¯ Now by Lemma 1.6, the sequence (kl)l∈N is in every component monotonely ¯ decreasing. It follows that the sequence (kl)l∈N eventually becomes constant. Thus we can assume that (T l, pl) = (T, p) for all l ∈ N for some structure tree ∼ ∼ n (T, p), and hence Hl = H = G for some H ∈ G . This proves the theorem. ¥

5This argument shows only the existence of an axiomatization, but is absolutely non- eﬀective. But at least theoretically, one could inductively determine the iterated Frattini quotients of the groups in G and this way receive an eﬀective axiomatization.

61 l ∼ l Example 3.54. In the case n = 1, the formula ϕl states that [Φ (G)] = Z/p ; the only pro-p-group fulﬁlling Φ = {ϕl | l ∈ N} is Zp.

3.5 Outlook

There are several other properties of absolute Galois groups one could check for axiomatizability: • The property stated in the Merkurjev-Suslin theorem: If K contains a primitive p-th root of unity and if [A] ∈ Br(K) is of order p, then A is equivalent to a product of cyclic algebras. Note that by Fact 1.32, this is a property of GK : 1 1 It is equivalent to say that the image of H (GK (p), Z/p ) × H (GK (p), Z/p ) 2 2 in H (GK (p), Z/p ) under the cup product generates H (GK (p), Z/p ). • Every field K which is not separably closed or pythagorean formally real has a proper extension L/K such that Gal(L/K) is projective, see [K5]. • There is a classification of the class of solvable absolute Galois groups, see [K6]. Is this subclass of the class of absolute Galois groups axiomatizable? • It rarely happens that an absolute Galois group GK decomposes as a direct product; if it does, then GK has to fulfill certain extra conditions, see [K7]. Can this property of absolute Galois groups be axiomatized? • For almost all fields K, the field K and the absolute Galois group GalK(t)/K 6 func of the rational function field over K (in an adequate language LIS ) are bi-interpretable, see [K1]. Can one use this result to axiomatize the class of absolute Galois groups GalK(t)/K of the rational function field over K for these fields K? Moreover, we have the following project: We know that absolute Galois groups (and thus the maximal abelian quotients of absolute Galois groups) fulfill the property (W) stated in Whaples’ theorem, cf. page 51. Using Theorem 5.25, we can easily construct proabelian groups G1,G2 with S(G1) ≺ S(G2) and such that G2 fulfills (W) and G1 does not: For Q n+1 Q n+1 example, choose G1 = n∈N Z/p and G2 = n∈N Z/p × Zp. We want to find an absolute Galois group G such that for the maximal abelian quotient ab ab ∼ G of G holds G = G2, and an elementary substructure S(H) ≺ S(G) with ab ∼ H = G1. Then it would be proven that the class AG is not axiomatizable.

6 I.e. the structure consisting of both the Galois groups GK(t) and GK and the restriction map between them.

62 Chapter 4

Demushkin groups

In this chapter, we provide the basic facts about Demushkin groups that we have used in Chapter 3, Section 3.4. There we discussed the “Elementary Type Conjecture” (Conjecture 3.40); Demushkin groups play a major role in the definition of the class G of groups occurring in the ETC. Moreover, we analyze the structure of Demushkin groups that are realized as Galois groups: It is conjectured that Demushkin Galois groups force a valuation structure on the field. It would be interesting in itself to find this valuation, but it might also help to prove the ETC. We hope that our analysis of Demushkin Galois groups will prove useful towards finding the conjectural valuation structure. The reader only interested in model theoretic questions might proceed to Chap- ter 5. In the whole chapter, we fix a prime p 6= 2. There is also a classification of 2-Demushkin groups similar to the one in Theorem 4.3, see [Lab1], Theorem 1; we restrict to the case p 6= 2.

4.1 Basics about Demushkin groups

We follow the presentation in [Lab1]. Let N≥1 := (N\{0})∪{∞} be as in Section 3.4. Deﬁnition 4.1. A pro-p-group G is called a Demushkin group if 1 1) dimFp H (G, Z/p ) < ∞, 2 2) dimFp H (G, Z/p ) = 1, 3) the cup product H1(G, Z/p )×H1(G, Z/p ) → H2(G, Z/p ) is non-degenerate. 1 Remark and Deﬁnition 4.2. 1) n(G) := dimFp H (G, Z/p ) equals the rank of G. 2) G is a one-relator pro-p-group, cf. Example 1.16.4: There is a short exact sequence 1 → R → F → G → 1, where F is a free pro-p-group of rank n and R = h|r|i is generated as a normal subgroup by one single element r ∈ F p [F,F ]. Passing to the maximal abelian

63 ab ab n quotient G , we obtain G as Zp modulo a subgroup either isomorphic to Zp or to zero. Thus ab ∼ i n−1 G = Z/p × Zp , ∞ where i ∈ N≥1 and by convention Z/p = 1. We set

q(G) := pi.

The next theorem shows that the invariants n(G) and q(G) determine G up to isomorphism. This was proven by S. Demushkin in [D]; a generalized and non-Russian proof can be found in [Lab1]. Theorem 4.3. 1) Let G be a Demushkin group of rank n(G) = n and with q(G) = q. Write G = F/h|r|i with F a free pro-p-group of rank n and r ∈ p F [F,F ] like above. Then n is even and there are generators x1, . . . , xn of F such that q r = x1[x1, x2][x3, x4] · ... · [xn−1, xn].

2) Let F be a free pro-p-group of rank n with generators x1, . . . , xn. For any q i r = x1[x1, x2][x3, x4] · ... · [xn−1, xn] with q = p for some i ∈ N≥1, the group F/h|r|i is a Demushkin group with invariants n(G) = n, q(G) = q.

It is easy to calculate the abelian height hp(G) of a Demushkin group. We use the following lemma: Q n li Lemma 4.4. Let G = i=1 Gi, where each Gi is a cyclic group of order p ∼ k with generator xi. Let N C G be a subgroup such that G/N = Z/p . Then there ∼ k+1 li is a subgroup M ≤ N such that G/M = Z/p if and only if ord (xiN) < p for all i ∈ {1, . . . , n}.

Proof: For “⇒”, note that in the canonical projection Z/p k+1 ³ Z/p k, the order of a preimage of a nontrivial element a is strictly bigger then ord(a). li For the other direction, assume that ord(xiN) < p for all i ∈ {1, . . . , n}. Then p li−1 we have xi ∈ N for all i ∈ {1, . . . , n}, thus G[p ] ⊆ N. Let ki be minimal such pki that xi ∈ N. As G/N is cyclic, there exists some i such that G/N = hxiNi; pk1 let us assume that G/N = hx1i. The element x1 is not p-divisible in N: If k k −1 p p 1 p 1 g = x1 , then g diﬀers only by an element from G[p ] from x1 , thus with pk1−1 p k1 g also x1 would be in N. Thus hx1 i splits as a direct factor of N; write k k +1 p 1 0 p 1 0 k+1 N = hx1 i × N . Let M := hx1 i × N ; then x1M has order p , and G/M =∼ Z/p k+1. ¥ Corollary 4.5. Let G be a Demushkin group with q(G) = pi. If N is an open normal subgroup such that G/N =∼ Z/p k, then there is an open normal subgroup ∼ k+1 i M of G, M ≤ N, such that G/M = Z/p if and only if ord(x1N) < p . In particular G has abelian height hp(G) = i.

Proof: This follows directly from the preceding lemma, as all occurring homo- k+i morphisms factorize through G/Gp [G, G] =∼ Z/p i × (Z/p k+i)n−1 if i 6= ∞, k+1 and through G/Gp [G, G] =∼ (Z/p k+1)n if i = ∞. ¥

64 × ∼ Remark and Deﬁnition 4.6. A homomorphism χ : G → Zp = Aut(Zp) turns i Zp into a G-module which we denote by I = I(χ). The quotients I/p I for i ≥ 1 are G-modules with the induced action of G.

The following theorem (which can be found in [Lab1]) will enable us in the next subsection to show that abelian height and cyclotomic height coincide for “properly” realized Demushkin groups. Theorem 4.7. Let G be a Demushkin group. Then there exists exactly one × homomorphism χ : G → Zp such that for I = I(χ) the canonical homomorphism 1 i 1 H (G, I/p I) → H (G, I/pI) is onto for all i ≥ 1. If x1, . . . , xn and q = q(G) q and r = x1[x1, x2][x3, x4]·...·[xn−1, xn] are as in Theorem 4.3, then χ is deﬁned by

−1 χ(x2) = (1 − q) and χ(xi) = 1 otherwise.

In particular we have im(χ) = 1 + qZp.

4.2 Demushkin groups as Galois groups

1 Definition 4.8. Let K be a field of characteristic 0 with ζp ∈ K. If GK (p) is a Demushkin group, we call K a Demushkin field. We abbreviate q(GK (p)) with q(K).

2 Examples for Demushkin fields are p-adic fields : If K ⊇ Qp is an extension of degree d such that the cyclotomic p-height of K is up(K) = i ≥ 1, then GK (p) is a Demushkin group of rank n(G) = d + 2 with q(G) = pi (cf. [Lab1], §5). Thus Demushkin groups which are realized over p-adic fields fulfill certain extra p−1 conditions connecting n = n(G) and q = q(G), one of them being n − 2 ≥ q p . With slight modifications, the proof shows that pup(K) = q(G) holds in arbitrary Demushkin fields. This is a well known result of Local Class Field Theory; short of a reference, we give a sketch of the proof:

up(K) Theorem 4.9. Let K be a Demushkin ﬁeld, let G = GK (p). Then p = q(G).

Sketch of proof: Let for n ∈ N∪{∞} be µpn = µpn (K(p)). Recall Deﬁnition and Example 3.33: The Galois action of G on µp∞ deﬁnes an homomorphism G → Aut(µp∞ /µp); after identifying Aut(µp∞ /µp) with 1 + pZp, this provides × an homomorphism χ : G → 1 + pZp ⊆ Zp . Let I = I(χ). Then for every n ∈ N, n the G-modules µpn and I/p I are isomorphic. n Using Fact 1.27 one gets for every n ∈ N a homomorphism K×/(K×)p =∼

1 Cf. [Ef]: There it is not required that ζp ∈ K – we include this condition in the definition in view of Example 4.21. 2In fact, the only known Demushkin fields are p-adically closed fields, i.e. fields elementarily equivalent to a p-adic field.

65 1 H (G, µpn ) given by

× pn a(K ) 7→ fa, p g( pn (a)) fa(g) = p pn (a)

× × × pn × × p for a ∈ K and g ∈ G. The canonical surjection K /(K ) ³√ K /(K ) pn 1 1 g( (a)) n √ induces a surjection H (G, µp ) ³ H (G, µp) given by [g 7→ pn ] 7→ [g 7→ √ (a) g( p (a)) √ ]. One gets a commutative diagram p (a)

× × pn ∼ 1 ∼ 1 n K /(K ) = H (G, µpn ) = H (G, I/p I)

× × p ∼ 1 ∼ 1 K /(K ) = H (G, µp) = H (G, I/pI)

Thus χ fulﬁlls the conditions from Theorem 4.7 and we have im(χ) = 1+q(G)Zp. But as χ corresponds to the Galois action of G on µp∞ , it follows that im(χ) = up(K) up(K) 1 + p Zp, and thus that p = q(G). ¥ Corollary 4.10. Let K be a Demushkin ﬁeld with q(K) = pi. Then i = hp(GK (p)) = up(K).

Proof: Clear with Corollary 4.5. ¥ The proof of Theorem 4.9 shows that if K is a Demushkin ﬁeld, then the homomorphism χ : G → 1 + pZp from Theorem 4.7 corresponds to the Galois action of G on µp∞ (K(p)). From this we get:

Corollary 4.11. Let K be a Demushkin ﬁeld and let G = GK (p), further let q = q(G). Let x1, . . . , xn be generators of G as in Theorem 4.3. For l ∈ N let l ζpl be a primitive p -th root of unity in K(p). Then

1) xj(ζpl ) = ζpl for j 6= 2 und thus K(µp∞ (K(p))) ⊆ Fix(h|x1, x3, . . . , xn|i). l l 2) x2(ζpl ) = ζpl for p ≤ q and x2(ζpl ) 6= ζpl for p > q.

l Proof: By the preceding proof, I(χ)/p I(χ) and µpl (K(p)) are for all l ∈ N isomorphic as G-modules. Now the corollary follows directly from Theorem 4.7. ¥

An absolute Galois group GK which is isomorphic to the absolute Galois group of a p-adic field is called a p-adic Galois group. As mentioned before, a field K is called p-adically closed if it is elementary equivalent to a p-adic field; this is equivalent to K having characteristic 0 and admitting a henselian valuation with finite residue field of characteristic p with a Z-group as value group, the value of p being finite (cf. [Pr]). One knows that an absolute Galois group GL for some field L is p-adic if and only if L is p-adically closed (cf. [K3]). Thus if the absolute Galois group is a p-adic group, then this forces a valuation structure on the field.

66 It is conjectured that if K contains a primitive p-th root of unity, then already the maximal pro-p quotient of the absolute Galois group of a p-adic field (a Demushkin group!) causes a valuation structure on K; and further that all Demushkin groups that can be realized as absolute pro-p Galois group of an arbitrary field are already realized over p-adic fields. We define: Definition 4.12. A Demushkin group that is realized as an absolute pro-p Ga- lois group of a p-adic fields is called a p-adic Demushkin group.

The precise conjecture is then:

Conjecture 4.13. Let K be a Demushkin field, in particular let G = GK (p) be a Demushkin group. 1) K has a valuation v with residue characteristic p and non p-divisible value group, whose decomposition field does not contain all roots of unity of p-power order. ∼ 2) G is p-adic Demushkin, i.e. there exists a p-adic field L with G = GL(p). In [Ef] it is shown that 1) implies 2); and that further the valuation from 1), if existing, has an especially nice form. How could one find a valuation structure on a Demushkin field? In a p-adic field × × p (K, v), if K(a)/K is inert, then NK(a)/K (K(a)) = Ov ·(K ) . Thus in this case, one can find the valuation ring on K if one knows which quotient of the Galois group corresponds to the inert extension. A first approach to find a valuation consists therefore in examining the realizations of p-adic Demushkin groups over p-adic fields in the hope that the inertia subgroup is uniformly3 definable. But this hope gets soon devastated: Corollary 4.11 shows that the fixed field of the normal subgroup T generated by x1, x3, . . . , xn is the cyclotomic extension L = K(µp∞ ). The extension L/K can either be totally ramified (if 1 − ζq is a uniformizer of K), or can contain a proper extension L0/K that is purely inert. Thus even if one considers only realizations of one fixed Demushkin group over p-adic fields, the quotient that belongs to the inertia subgroup can vary. The next attempt is to use the following fact: A field K admits a valuation v with non p-divisible value group if and only if there is a subgroup S ≤ K× of index p such that S + aS ⊆ S ∪ aS for all a ∈ K× \ S, cf. [K4], Corollary 3.2. To be able to find a valuation structure by specifying such a subgroup S, we need to understand the maximal multiplicative subgroups of Demushkin fields. In the following, we will examine two candidates for the desired maximal subgroup; but we will show that the two subgroups are equal, and do not satisfy the necessary conditions. Recall that L/K is a pi-dead end if L/K is cyclic of degree pi and there is no cyclic extension M/L such that M/K is cyclic of degree pi+1, cf. Definition 3.21. Lemma 4.14. Let K be a Demushkin field with q(K) = pi, let a ∈ K× \(K×)p. Let x1, . . . , xn be generators of GK (p) as in Theorem 4.3. Then √ √ √ pi i p p K( a)/K is a p -dead end ⇐⇒ x1( a) 6= a.

3Uniformly here means uniformly for all p-adic Demushkin groups, for example in terms of generators of the subgroup depending on the usual generators x1, . . . , xn, and not only uniformly for the diﬀerent realizations of one particular p-adic Demushkin group G.

67 Proof: We have √ K( pi a)/K is a pi-dead end ⇐⇒4.5 x has order pi 1| √i K( p a) ³ √ ´ ⇐⇒ x generates Gal K( pi a)/K 1| √i K( p a)

68 ⇐⇒ Fix(x ) = K 1| √i K( p a) √ √ p p ⇐⇒ x1( a) 6= a. ¥ The lemma justifies the following definition: Definition 4.15. Let K be a Demushkin field with q(K) = pi. √ 1) An element a(K×)p ∈ K/(K×)p is a dead end if the extension K( pi a)/K i is a p -dead end. √ 2) An element a(K×)p ∈ K/(K×)p is a through road if the extension K( pi a)/K is no dead end, or if a(K×)p = (K×)p. 3) We also call elements of K× dead ends or through roads, depending on whether their residues modulo (K×)p are dead ends or through roads.

As the following lemma shows, the through roads constitute a subgroup of index p in K×/(K×)p, and thus yield a candidate for the maximal subgroup S in demand: Lemma 4.16. Let K be a Demushkin ﬁeld. Then the set T of through roads is a subgroup of index p in K×/(K×)p.

Proof: It follows directly from the Lemma 4.14 that T is a subgroup. The cosets of T are the sets √ √ × p × p p p {a(K ) ∈ K/(K ) | x1( a) = ζ a } for some p-th root of unity ζ, thus T has p cosets. ¥

Lemma 4.17. Let V be a vector space of dimension n over Fp. Then 1) The number u(m) of subspaces of dimension m of V is

mY−1 pn − pk u(m) = . pm − pk k=0

pn−1 2) In particular, it follows that u(1) = u(n − 1) = p−1 . Proof: The proof is an easy calculation and can be found in basic textbooks about linear algebra. ¥ Lemma 4.18. Let K be a ﬁeld containing a primitive p-th root of unity, let G = GK (p). Then n = n(G) equals the dimension r of the Fp-vector space K×/(K×)p.

Proof: Let u be the number of Z/p -extensions of K. By the main theorem of Galois theory, the Z/p -extensions of K correspond to subgroups of index p in G/Φ(G), which are just the subspaces of codimension 1 in G/Φ(G) considered pn−1 as an Fp-vector space. Thus by the last lemma, we have u = p−1 . By Lemma 1.28, the number u also equals the number of cyclic subgroups of order p in K×/(K×)p, which are just the subspaces of dimension 1 in K×/(K×)p pr −1 considered as an Fp-vector space. By the last lemma thus u = p−1 , hence we have r = n. ¥

69 The other subgroup of index p in K× we consider is the subgroup consisting of the cyclotomic norms Nζq(G) . This is in fact a maximal subgroup; more general we have: p Lemma 4.19. Let K be a Demushkin ﬁeld, let a ∈ K \ K . Then Na is a subgroup of index p in K×.

× Proof: Consider (a, ): K → pBr(K), b 7→ (a, b), where (a, b) = (a, b, ζp) is the norm rest algebra determined by a and b relative to a ﬁxed primitive 1 p-th root of unity ζp. Recall Remark 1.33. As the cup product H (G, Z/p ) × H1(G, Z/p ) → H2(G, Z/p ) =∼ Z/p is non-degenerate by the deﬁnition of a Demushkin group, the map (a, ) is surjective; further by Fact 1.25 we have × ∼ ∼ ker((a, )) = Na, so K /Na = pBr(K) = Z/p. ¥

In the next theorem we show that the subgroup Nζq(G) of cyclotomic norms and the subgroup of through roads coincide. Theorem 4.20. Let K be a Demushkin ﬁeld, let q = q(K) and let a ∈ K×. Let ζ be a primitive q-th root of unity in K. Then × p a(K ) is a through road ⇐⇒ a ∈ Nζ .

Proof: Let G = GK (p) and n = n(G), let x1, . . . , xn be generators of G as in Theorem 4.3. For notational convenience, we prove the theorem only in the case q = p; the proof for other q-invariants works the same way. Let T ⊆ K×/(K×)p be the subgroup of through roads. Fix a p-th root α of a and a primitive p2-th p root of unity ζp2 with (ζp2 ) = ζ. We have × p a(K ) is a through road ⇐⇒ x1(α) = α and

a ∈ Nζ ⇐⇒ (a, ζ, ζ) ∼ 1 ⇐⇒ fa,ζ is a coboundary, where fa,ζ is the cocycle deﬁned in Remark 1.33: 2 fa,ζ : G → Z/p, fa,ζ (x, y) = ij,

y(ζ ) where i, j are deﬁned by the equations x(α) = ζi and p2 = ζj. α ζp2 × p Both the through roads as well as N(ζp) · (K ) are maximal subgroups of × × p × p K /(K ) , thus we are done if we can show that Nζ · (K ) ⊆ T . Assume not; × p then there is a ∈ Nζ such that a(K ) is a dead end. By the above, this means that fa,ζ is a coboundary, but x1(α) 6= α. We will derive a contradiction from this assumption. By choosing the primitive p-th root of unity ζ properly (note that the norm x1(α) group Nζ does not depend on this choice), we can assume that α = ζ. Let x2(ζ 2 ) x2(α) j1 p j2 f = fa,ζ . Further let = ζ und = ζ (recall that ζ ∈ K). From α ζp2 Corollary 4.11 it follows that j2 6= 0. As f is a coboundary, there exists a continuous map u : G → Z/p such that f(x, y) = u(x) + u(y) − u(xy) for all x, y ∈ G. Let u(xi) = ui and let id be the identity in G. We use the notation log(ζi) = i ∈ Z/p for every power of ζ; with this notation we have y(ζ ) f(x, y) = log( x(α) )log( p2 ). α ζp2

70 Claim 1: u(id) = 0 id(ζ ) Proof of Claim 1: f(id, id) = log( id(α) )log( p2 ) = log(ζ0)log(ζ0) = 0 = α ζp2 u(id) + u(id) − u(id), thus u(id) = 0. k Claim 2: For i 6= 2 and k ∈ Z we have u(xi ) = kui. Proof of Claim 2: For k = 0 this is the statement of Claim 1. • Induction for k ≥ 0: Assume the result is proven for all l with 0 ≤ l ≤ k. Recall that by Corollary 4.11, we have xi(ζp2 ) = ζp2 . Then k x (ζ ) k k xi (α) i p2 xi (α) k+1 f(xi , xi) = log( )log( ) = log( ) ·0 = kui +ui −u(xi ), thus α ζp2 α k+1 u(xi ) = (k + 1)ui. • For k = −1 : −1 −1 xi(α) xi (ζp2 ) −1 −1 f(xi, xi ) = log( )log( ) = 0 = ui+u(xi )−u(id) = ui+u(xi ), α ζp2 −1 thus u(xi ) = −ui. • Induction for k < 0: Assume the result is proven for all l with 0 ≥ l ≥ k. Then k x−1(ζ ) k −1 xi (α) i p2 k−1 k−1 f(xi , xi ) = log( )log( ) = 0 = kui −ui −u(xi ), thus u(xi ) α ζp2 = (k − 1)ui.

Claim 3: u([x1, x2]) = −j2 Proof of Claim 3: x1(α) x2(ζp2 ) • f(x1, x2) = log( )log( ) = j2 = u1 + u2 − u(x1x2), thus u(x1x2) = α ζp2 u1 + u2 − j2. −1 x (ζ 2 ) −1 −j2 −1 x2(α) 2 p • Note that x2 (ζp2 ) = ζ ζp2 . Thus f(x2, x2 ) = log( )log( ) = α ζp2 −1 −1 −j1j2 = u2 + u(x2 ) − 0, hence u(x2 ) = −j1j2 − u2. −1 x−1(ζ ) −1 −1 x1 (α) 2 p2 • f(x1 , x2 ) = log( )log( ) = (−1)(−j2) = j2 = −u1 − u2 − α ζp2 −1 −1 −1 −1 j1j2 − u(x1 x2 ), thus u(x1 x2 ) = −u1 − u2 − j2 − j1j2. −1 −1 x x (ζ ) −1 −1 x1 x2 (α) 1 2 p2 −1 −1 • f(x1 x2 , x1x2) = log( )log( ) = (−1 − j1)j2 = u(x1 x2 ) α ζp2 + u(x1x2) − u([x1, x2]) = −u1 − u2 − j2 − j1j2 + u1 + u2 − j2 − u([x1, x2]), hence u([x1, x2]) = −j2.

Claim 4: u([xk, xl]) = 0 for k, l 6= 2. Proof of Claim 4: xk(α) xl(ζp2 ) xk(α) • f(xk, xl) = log( )log( ) = log( ) · 0 = 0 = uk + ul − u(xkxl), α ζp2 α hence u(xkxl) = uk + ul. −1 −1 • Analogously u(xk xl ) = −uk − ul. −1 −1 x x (ζ ) −1 −1 xk xl (α) k l p2 • f(x x , xkxl) = log( )log( ) = 0 = u([xk, xl]). k l α ζp2

Claim 5: u([x3, x4] · ... · [xk−1, xk]) = 0 for k ≤ n and k even. Proof of Claim 5: For k = 4 this follows from Claim 4. Induction over k: f([x3, x4] · ... · [xk−1, xk], [xk+1, xk+2]) = −u([x3, x4] · ... · [xk+1, xk+2]) = 0.

Claim 6: u([x1, x2] · ... · [xn−1, xn]) = −j2.

Proof of Claim 6: f([x1, x2], [x3, x4] · ... · [xn−1, xn]) = 0 = u([x1, x2]) − u([x1, x2] · ... · [xn−1, xn]) = −j2 − u([x1, x2] · ... · [xn−1, xn]), hence u([x1, x2] ·

71 ... · [xn−1, xn]) = −j2. p Now consider f(x1, [x1, x2]·...·[xn−1, xn]). By the definition of u and the above we have f(xp, [x , x ] · ... · [x , x ]) = pu − j + u(xp[x , x ] · ... · [x , x ]) 1 1 2 n−1 n 1 2 | 1 1 2 {z n−1 n } =u(id)=0 p But x1(α) = α, thus by the definition of f we have µ ¶ p [x1, x2] · ... · [xn−1, xn](ζp2 ) f(x1, [x1, x2] · ... · [xn−1, xn]) = 0 · log = 0, ζp2 and hence 0 = j2. This is a contradiction, because j2 6= 0 as we already noted before ! ¥ The theorem shows that the subgroup T of through roads cannot be the subgroup that allows to define the valuation structure: It is well known that in × × p × p p-adic fields, S = Ov · (K ) is the only subgroup of index p containing (K ) such that S + aS ⊆ S ∪ aS for all a ∈ K× \ S. We have already seen on page 67 that N = T is in general not equal to S. ζpi This result is not very astonishing, as T has a group theoretic definition by Lemma 4.14; the considerations of page 67 have shown that a uniform group theoretic characterization of S is not possible. A next possible approach to find S could be to use a combinatorial construction like in [K4]. This attempt is difficult because even the simplest example q(G) = p = 3 and n(G) = 4, which corresponds to the 3-Galois group of Q3(ζ3), has a high combinatorial complexity and can hardly be handled even when using a computer algebra systems for the computations. We finish this subsection with an example that shows that Demushkin groups can also be realized without a primitive p-th root of unity in the ground field. The idea is to consider an extension of Q which does not contain a primitive p-th root of unity ζ, but an element ε which is p-adically close to ζ. As the example is not central for our considerations, we will not carry out all valuation theoretic arguments in detail. Example 4.21. For every prime p > 2, there is a field K ⊇ Q not containing a primitive p-th root of unity ζp such that GK (p) is a Demushkin group of rank p + 1 with q(GK (p)) = 1.

Sketch of proof (See Picture 4.21): Fix some extension v of the p-adic val- a p p p uation on Qp to Qp. Let f(X) = X − 1 and let g(X) = X − (1 + p ). Let ε1, . . . , εp be the zeros of g. Then Yp p (ζp − εi) = g(ζp) = (g − f)(ζp) = −p , i=1 thus v(ζp − εi) ≥ 1 for some ε = εi. Now consider the minimal polynomial p−1 p−2 Qp−1 i h(X) = X + X + ··· + X + 1 = i=1 (X − ζp) of ζp over Qp; considering i 1 h(1), we see v(1 − ζp) = p−1 . Further we have for 2 ≤ j ≤ p − 1 that 1 1 v(ζ − ζj) = v(ζ ) + v(1 − ζj−1) = 0 + = , p p p p p − 1 p − 1

72 and with Krasner’s Lemma (see Theorem 4.1.7 in [EnP]) this shows that Qp(ζp) ⊆ Qp(ε). As [Qp(ζp): Qp] = p − 1 and [Qp(ε): Qp] ≤ p, it follows that Qp(ζp) = Qp(ε) =: L.

One easily sees that [Q(ε): Q] = p and hence ζp 6∈ Q(ε). Note that L ist the completion of Q(ε). h,p Let K = Q(ε) ⊆ Qp(ε) be a p-henselization of Q(ε), cf. Section 4.2 in [EnP]. Then K/Q(ε) is a p-extension, thus does not contain ζp. Further L is the completion of K.

K(p) L := Qp(ε) = Qp(ζp) p − 1 dense, immediate Qp

K := Q(ε)h,p

dense, Q(ε) immediate p Q

Picture 4.21

L/K is an immediate extension, K(p)/K is a Galois p-extension. If x ∈ K(p)∩L, then K(x)/K is an immediate p-extension. But by Theorem 4.1.10 in [EnP]4, K has no proper immediate p-extensions, thus x ∈ K. As K(p)/K is Galois, it follows that K(p) and L are linearly disjoint over K.

We have [L : Qp] = p − 1 and up(L) = 1, thus GL(p) = G for a Demushkin group of rank p + 1 and with q(G) = p. So we are done when we can show that the restriction GL(p) ³ GK (p) is bijective. • The surjectivity follows directly from the linear disjointness of K(p) and L over K. • Let 1 6= σ ∈ GL(p), say σ(α) 6= α for some α ∈ L(p). As K is dense in its completion L, we can use Krasner’s Lemma to ﬁnd β ∈ K(p) such that L(β) = L(α); thus σ(β) 6= β, and hence σ 6= idK(p). |K(p) ¥

4.3 Demushkin groups as semidirect products

It is well known and can be shown with valuation theory and local class field theory that a Demushkin group which is realized as an absolute pro-p-Galois group of a p-adic field has a presentation as a semidirect product ZpoH, where H is a closed normal subgroup that is free pro-p of countable infinite rank. We show

4The proof there is for henselian ﬁelds, but translates directly to the p-Hensel case.

73 that such a decomposition exists for arbitrary Demushkin groups. We use a well- known lemma that holds more generally for Poincarégroups: A Poincarégroup of dimension n is a finitely generated pro-p-group such that Hn(G, Z/p ) =∼ Z/p , Hi(G, Z/p ) is finite for all i ≥ 1, and such that the cup-product induces a non- degenerate pairing Hi(G, Z/p )×Hn−i(G, Z/p ) → Hn(G, Z/p ) for all 0 ≤ i ≤ n. Demushkin groups are just Poincarégroups of dimension 2. The lemma is the statement of Exercise 3 in Chapter III of [NSW]; see also page 273 in [Wi]. Lemma 4.22. Let G be a Poincarégroup of dimension n > 0. If H ≤ G is a closed, but not open subgroup, then cd(H) ≤ n − 1. Theorem 4.23. Let G be a Demushkin group of rank n(G) = n with q(G) = pi. ∼ If H is a closed normal subgroup such that G/H = Zp, then H is free pro-p of ∼ countable infinite rank and G = H o Zp. In particular it follows that every Demushkin group decomposes as such a semidirect product.

Proof: It is clear that such closed normal subgroups H exist; for example, take H = h|x1, x2, . . . , xn−1|i in the usual presentation of G. Now let G and H be like in the theorem. The last lemma shows that H has cohomological dimension 1; thus H is free pro-p. Being a closed subgroup of a ﬁnitely generated group, H is countably generated. Further, as Zp is free pro-p and thus projective, the sequence 1 → H → G → Zp → 1 splits and hence G decomposes as a semidirect product G = H o Zp. It remains to show that H has inﬁnite rank. Consider the descending chain of subgroups

G := G =∼ Z o H ≥ pZ o H ≥ p2Z o H ≥ .... 0 | p {z } | p{z } | p{z } :=G1 :=G2 :=G3

i One easily checks that [Gi+1 : Gi] = p, thus ni := rank(Gi) = p (n − 2) + 2, cf. Theorem 3.9.15 in [NSW]. As obviously rank(Gi) ≤ 1 + rank(H), it follows that rank(H) ≥ ni − 1 for all i ∈ N, hence H has countably inﬁnite rank. ¥

4.4 An alternative deﬁnition of Demushkin Galois groups

Theorem 4.24. Let K be a ﬁeld such that G = GK (p) is a ﬁnitely generated pro-p-group, let ζp ∈ K be a primitive p-th root of unity. Then the following are equivalent: 1) G is a Demushkin group. 2) The map

ϕ : {Z/p-extensions of K} → {U ≤ K×/(K×)p | U has index p},

× p Ka/K 7→ Na · (K ) , is bijective.

74 Proof: In the following proof, the symbol (a, b) does by abuse of notation both denote the norm rest algebra (a, b, ζp), and the equivalence class of (a, b) in pBr(K). 1) ⇒ 2): Let G be a Demushkin group, let n = n(G). Lemma 4.19 shows that ϕ is well defined. ¯ ¯ ¯ ¯ Claim: ¯{Z/p-extensions of K}¯ = ¯{U ≤ K×/(K×)p | U has index p}¯. From the proof of Lemma 4.18 we know that the number of Z/p- extensions of K equals the number of subspaces of dimension 1 in the Fp-vector space K×/(K×)p, and thus is because of Lemma 4.17 equal to the number of subspaces of codimension 1. The subspaces of codimension 1 in K×/(K×)p correspond to subgroups of index p in K×/(K×)p. × × p By¯ Lemma 4.18, the dimension of K¯ /(K ) as Fp-vector space equals n. Thus ¯{U ≤ K×/(K×)p | U has index p}¯ is finite, and it suffices to show that ϕ is 0 × injective. We assume the contrary: Let Na = Na0 for some a, a ∈ K with × p 0 × p Ka 6= Ka0 . Then using Lemma 1.28, we see a(K ) 6= a (K ) ; moreover for all b ∈ K× we have (a, b) ∼ K ⇐⇒ (a0, b) ∼ K. Claim: There exists i ∈ N such that (ai, y) ∼ (a0, y) holds for all y ∈ K×. For y ∈ K× with (a, y) ∼ K this holds for arbitrary i. Choose x ∈ K× such that (a, x) 6∼ K. Then (a, x) is a generator of pBr(K) =∼ H2(G, Z/p ) =∼ Z/p , thus there is i ∈ {0, . . . , p − 1} such that (a, x)i ∼ (ai, x) ∼ (a0, x). We show that (ai, y) ∼ (a0, y) for arbitrary y ∈ K×: Let y ∈ K×. Choose l ∈ {0, . . . , p − 1} such that (a, x)l = (a, y). Then l −1 l −1 (a, x y ) = 1, thus x y ∈ Na = Na0 ; this means that there is n ∈ Na = Na0 with y = nxl. Hence we have (ai, y) ∼ (ai, xl) ∼ (ai, x)l ∼ (a0, x)l ∼ (a0, xl) ∼ (a0, y). This proves the claim. It follows that (a−ia0, y) ∼ K for all y ∈ K×. As the cup product H1(G, Z/p ) × H1(G, Z/p ) → H2(G, Z/p ) is non-degenerate and by Remark 1.33, it follows that a−ia0 ∈ (K×)p, hence ai(K×)p = a0(K×)p. Finally with Lemma 4.19 we get Ka = Ka0 , a contradiction. This proves 1) ⇒ 2). 1 2) ⇒ 1): As G is finitely generated, we have dimFp H (G, Z/p) < ∞. By the × assumptions, the norm group Na is a proper subgroup of K for every a ∈ × × p K \ (K ) . Then for 0 6= b 6∈ Na we have (a, b) 6∼ K; by Remark 1.33, this shows that the cup product H1(G, Z/p ) × H1(G, Z/p ) → H2(G, Z/p ) is non- degenerate. It remains to show that H2(G, Z/p) =∼ Z/p. Choose a, b ∈ K× with (a, b) 6∼ K, and let (c, d) be arbitrary. We show that (c, d) ∈ h(a, b)i.

Claim: If at least one of the extensions Kc/K, Kd/K is equal to either Ka/K or Kb/K, then (c, d) ∈ h(a, b)i.

Assume Kc/K equals Ka/K (the other cases are similar). By Lemma 1.28, we have c = alkp for some k ∈ K× and 0 ≤ l ≤ p − 1. As (a, b) 6∼ K, i × j the cosets b Na of Na for 0 ≤ i ≤ p − 1 cover K ; let d ∈ b Na. Then −j l p −j l p l p −j db ∈ Na, thus K ∼ (a k , db ) ∼ (a k , d)(a k , b ), which implies that (c, d) ∼ (alkp, d) ∼ (alkp, bj) ∈ h(a, b)i. Now let c, d be arbitrary; we can assume that (c, d) 6∼ K. Choose an arbitrary x ∈ × K \ (Nb ∪ Nc). Then K 6∼ (x, b) and K 6∼ (x, c); using the claim once for (x, b) and (a, b) and once for (x, c) and (c, d), we get (x, b) ∼ (a, b)r and (x, c) ∼ (c, d)s

75 for some 1 ≤ r, s ≤ p − 1. Thus h(a, b)i = h(x, b)i and h(c, d)i = h(x, c)i. But using the claim again, we also get h(x, b)i = h(x, c)i and thus h(a, b)i = h(c, d)i, which ﬁnishes the proof. ¥

76 Chapter 5

A classiﬁcation of proabelian groups

Notations and Conventions: As usual when dealing with abelian groups, we will use additive notation. We will denote abstract abelian groups by A, B,..., while A, B, . . . is reserved for proabelian groups. Further if A, B are torsion groups, then A∗,B∗, A∗, B∗ denote the Pontryagin duals of A, B, A, B respectively. When we write S(A) ≡ S(B), we mean elementary equivalence of the complete inverse systems in LIS, whereas A ≡ B or A ≡ B is used for elementary equivalence in the group language LG. Nevertheless, we will sometimes especially emphasize the current language. As in Subsection 1.4.2, let lv(κ) be the logical value of a cardinal κ. Let N := N ∪ {∞}; we call a sequence (xn)n∈N an N-sequence. The xn will in the sequel be either cardinals or logical values of cardinals. We will extensively make use of the following notation: If (xn)n∈N and x are given, we denote the N-sequence (xn)n∈N with x∞ = x by ((xn)n∈N, x). Deﬁnition 5.1.

1) Let U = (un)n∈N be an N-sequence of cardinal numbers. If (un)n∈N eventually becomes zero, we say that U is finite, otherwise we call it infinite. Analogously we define finite and infinite N-sequences of logical values of cardinals. 0 0 2) We say that two N-sequences U = (un)n∈N, U = (un)n∈N are equivalent, 0 if either both sequences are infinite and lv(un) = lv(un) for all n ∈ N, or 0 if both systems are finite and lv(un) = lv(un) for all n ∈ N. Analogously 0 we define equivalence of N-sequences (lv(un))n∈N, (lv(un))n∈N. We denote the equivalence class of U by U.

If the meaning is clear from the context, we will sometimes not mention explicitly whether an N-sequence consists of cardinal numbers or of logical values of cardinal numbers. Content and structure of Chapter 5: In Section 5.1, we classify proabelian groups up to elementary equivalence in LIS and give invariants for every complete LIS-theory of proabelian groups. The main theorem that we prove in this section is

77 Theorem 5.2. (First version) Let A be proabelian. Then there is some direct product Y Y n+1 up,n Divp B := (Z/p ) × Zp . n∈N, p prime p prime such that S(A) is elementarily equivalent to S(B). The complete inverse systems of two such products are elementarily equivalent if and only if the N-sequences ((up,n)n∈N, Divp) are equivalent for all primes p.

There will be a sharper version of the theorem on page 90, where we will show how the N-sequences ((up,n)n∈N, Divp) can be calculated from a given proabelian group A. In Section 5.2, we show that the complete inverse systems S(A),S(B) of two proabelian groups A, B are LIS-elementarily equivalent if and only if the Pon- ∗ ∗ tryagin duals A ,B of A, B are LG-elementarily equivalent (Theorem 5.22). ∗ ∗ Moreover, we show that A, B are LG-elementarily equivalent if and only if A ,B are LG-elementarily equivalent (Theorem 5.26). All together, we get that A, B are LIS-elementarily equivalent if and only if A, B are LG-elementarily equivalent. Finally, in Section 5.3, we state some results about categoricity, stability and the structure of saturated models.

5.1 A ﬁrst classiﬁcation

Before we start proving Theorem 5.2, we consider some examples. First note that not every proabelian A groups is a direct product of procyclic groups: Remark 5.3. A proabelian group A is a direct product of procyclic groups if and only if the Pontryagin dual A∗ decomposes into a direct sum of cyclic groups and Prufer¨ groups. Example 5.4. • Let D be the discrete abelian torsion group with generators x, y1, y2,... subject 2 to relations 0 = px, x = py1 = p y2 = .... Then D does not decompose as a direct sum of cyclic groups and Prufer¨ groups, see Example 5.15. Thus the dual group D := D∗ is not a direct product of procyclic groups. In fact, one can show that Y n+1 D := { (xn)n∈N ∈ (Z/p ) | xn ≡ x1 mod p}. n∈N

Q n+1 • The complete inverse systems belonging to the groups P := n∈N(Z/p ), P × Zp and D from above are elementarily equivalent, but pairwise not isomorphic. ℵ0 ℵ0 2 • S((Z/p ) × Zp) 6≡ S((Z/p ) × Zp).

Proof: We will prove in Example 5.15 that D does not decompose as a direct ∼ ∼ product of procyclic groups; it follows that D =6 P and D =6 P ×Zp, and thus the corresponding complete inverse systems are not isomorphic as well. Further in

78 Example 5.15 it is shown that S(D) ≡ S(P ). The statements S(P ) ≡ S(P ×Zp) ℵ0 ℵ0 2 and S((Z/p ) × Zp) 6≡ S((Z/p ) × Zp) follow directly from Theorem 5.2. The ∼ result P =6 P × Zp is shown in [C1]; compare also Observation 5.30. ¥ The ﬁrst step to prove Theorem 5.2 is to show that for proabelian groups A, B of ﬁnite exponent, we have

S(A) ≡ S(B) ⇐⇒ A∗ ≡ B∗; this will be done in Subsection 5.1.1, Theorem 5.7. In Subsection 5.1.2, we use this result to prove Theorem 5.2.

5.1.1 Elementary equivalence of proabelian groups of ﬁnite exponent

To prove that S(A) ≡ S(B) ⇐⇒ A∗ ≡ B∗ holds for proabelian groups A, B of finite exponent, we define appropriate Ehren- feucht-Fraisségames for abelian groups of finite exponent and then use Pon- tryagin duality and the characterization of elementary equivalence of complete inverse systems from Section 2.3. The following Ehrenfeucht-Fraisségame is adapted to abelian groups of finite exponent. The idea is that the players do not play with single elements, but with subgroups of bounded size: In the first move, the players choose subgroups generated by one element, and isomorphisms between them. In the second move, they choose subgroups generated by the subgroup from the first move and one additional element, and so on.

Definition 5.5. Let A, B be abelian groups of finite exponent, let C0 be a common m,s finite subgroup of cardinality r. Let m, s ≥ 1. The game Gbd (AC0 , BC0 ) has the following rules:

• In the ﬁrst move, player I chooses a subgroup A1 ≤ A with C0 ≤ A1 and |A1| ≤ rs, or a subgroup B1 ≤ B with C0 ≤ B1 and |B1| ≤ rs. Player II then ∼ chooses B1 ≤ B (respectively A1 ≤ A) and an isomorphism ϕ1 : A1 = B1 such

that ϕ1 = idC0 . |C0 i • In the i-th move, player I chooses a subgroup Ai ≤ A with |Ai| ≤ rs and such i that Ai ≥ Ai−1, or a subgroup Bi ≤ B with |Bi| ≤ rs and such that Bi ≥ Bi−1. Player II has to choose Bi ≤ B (respectively Ai ≤ A) and an isomorphism ∼ ϕi : Ai = Bi such that ϕi = ϕi−1. |Ai−1 • Player II wins the game if and only if he can find for all i ≤ m a subgroup ∼ Bi ≤ B (respectively Ai ≤ A) and ϕi : Ai = Bi as requested. m,s We write Gbd (A, B) for the game over the trivial subgroup 0. The next lemma states that for abelian groups of finite exponent the above game is equivalent to the usual Ehrenfeucht-Fraisségame. The proof is easy, but technical. It is crucial that the groups have finite exponent, as this guarantees the existence of a bound for the cardinalities of the subgroups.

79 Lemma 5.6. Let A, B be abelian groups of finite exponent, say nA = 0 and nB = 0. Let C be a common subgroup. Then AC ≡ BC if and only if for all m, s ≥ 1 and all finite subgroups C0 ≤ C, there are winning strategies for the m,s games Gbd (AC0 , BC0 ), if and only if for all m ≥ 1 and all finite subgroups C0 ≤ C, m,n there are winning strategies for the games Gbd (AC0 , BC0 ). In particular, A ≡ B if and only if there are winning strategies for the games m,s Gbd (A, B).

Proof: Assume that for all m ≥ 1 and all finite subgroups C0 ≤ C, there are m,n winning strategies for the games Gbd (AC0 , BC0 ). We want to prove that AC ≡ BC. For this purpose, it suffices to show that for all m ≥ 1 and all finite tuples c¯ ∈ C, m the usual Ehrenfeucht games G (Ac¯, Bc¯) of length m with parameters c¯ have winning strategies. k Let m, k ≥ 1 and let c¯ = (c1, . . . , ck) ∈ C be arbitrary. Let C0 = hc1, . . . , cki and k m,n let r = |C0| ≤ n . We use the winning strategy W for the game Gbd (AC0 , BC0 ) m to find a winning strategy for the usual Ehrenfeucht game G (Ac¯, Bc¯) of length m with parameters c¯: Assume that in the first move, player I chooses an element a1 ∈ A. Then A1 = ha1, C0i is a subgroup of cardinality at most r · n of A. Player II uses W to ∼ choose B1 ≤ B and ϕ1 : A1 = B1 such that ϕ1 = idC0 . He picks the element |C0 b1 := ϕ1(a1). The case that player I starts with b1 ∈ B is analogous. Assume now that in the i-th move, player I chooses an element ai ∈ A. Let i−1 i Ai = hAi−1, aii; as inductively |Ai−1| ≤ r · n , we have |Ai| ≤ r · n . Player II ∼ uses W again to choose Bi ≤ B and ϕi : Ai = Bi extending ϕi−1. Player II picks bi := ϕi(ai); again, the case that player I starts with bi ∈ B is similar. After m ∼ moves, the isomorphism ϕm : Am = Bm, which is the identity on c¯, restricts to a partial LG ∪ {c¯}-isomorphism {ai 7→ bi | 1 ≤ i ≤ m}. Thus player II has a m winning strategy for the usual Ehrenfeucht game G (Ac¯, Bc¯) of length m with parameters c¯.

Now assume that AC ≡ BC. This implies that for all m ≥ 1 and all finite tuples m c¯ ∈ C, the usual Ehrenfeucht game G (Ac¯, Bc¯) of length m with parameters c¯ has a winning strategy. We show that for all m, s ≥ 1 and all finite subgroups m,s C0 ≤ C, there are winning strategies for the games Gbd (AC0 , BC0 ). Let m, s ≥ 1 be arbitrary, let C0 ≤ C be an arbitrary finite subgroup, say C0 = hc1, . . . , cki and |C0| = r. Let c¯ = (c1, . . . , ck). Let W be a winning rsm strategy for the game G (Ac¯, Bc¯); we deduce a winning strategy for the game m,s Gbd (AC0 , BC0 ).

Assume player I chooses in the ﬁrst move A1 = {a1, . . . aj1 } ≤ A, A1 ≥ C0 with j1 ≤ r · s. Then player II uses j1-times the strategy W to ﬁnd b1, . . . bj1 such ϕ1 that aj 7→ bj is a partial LG ∪ {c¯}-isomorphism; thus ϕ1 is the identity on C0.

Then B1 = {b1, . . . , bj1 } is a subgroup of B with C0 ≤ B1. i If in the i-th move player I chooses Ai = {a1, . . . aji } ≤ A with ji ≤ r · s and

Ai ≥ Ai−1 = {a1, . . . aji−1 }, then player II uses (ji−ji−1)-times the strategy W to

ﬁnd bji−1+1, . . . bji corresponding to the aji−1+1, . . . , aji ; again Bi := {b1, . . . , bji } ϕi is a subgroup of B and aj 7→ bj is a partial LG ∪{c¯}-isomorphism. After m moves, the Ai, Bi and ϕi for 1 ≤ i ≤ m are as required. Thus player II has found a win- m,s ning strategy for Gbd (AC0 , BC0 ). ¥ Now the announced result follows easily with the results in Section 2.3 and

80 Pontryagin duality: Theorem 5.7. Let A, B be proabelian groups of ﬁnite exponent.Let S(C) be a common substructure of S(A) and S(B); we can consider C∗ as a common subgroup of A∗ and B∗. Then

∗ ∗ S(A)S(C) ≡ S(B)S(C) ⇐⇒ AC∗ ≡ BC∗ . In particular S(A) ≡ S(B) ⇐⇒ A∗ ≡ B∗.

Proof: Let n ≥ 1 such that nA = 0 and nB = 0. Then we also have nA∗ = 0 and nB∗ = 0, see Corollary 1.15. Let [c] ∈ S(C)/∼ be an arbitrary equivalence class. Then [c]∗ can be considered as a subgroup of C∗, and every ﬁnite subgroup ∗ ∗ C0 ≤ C has the form [c] for some c ∈ S(C). Further we know that chains

[Nm] J³ ... J³ [N1] J³ [c] of equivalence classes in S(A) or S(B) correspond to chains

∗ ∗ ∗ [c] ≤ N1 ≤ ... ≤ Nm

∗ ∗ ∗ ∗ of subgroups in A or B , the indices [[Ni+1]:[Ni]] = [Ni+1 : Ni ] being the same.

Let S(A)S(C) ≡ S(B)S(C). By Lemma 2.9, this is equivalent to the property m,s that for all m, s ≥ 1 and for every c ∈ Ss(C) the game GIS (S(A)c,S(B)c) ∗ ∗ has a winning strategy. Fix m ≥ 1, and a finite subgroup C0 = [c] ≤ C , let r = max{n, |C0|}. Then c ∈ Sr(C). By dualizing and using a winning strategy m,r m,n ∗ ∗ for the game GIS (S(A)c,S(B)c), player II can win the game Gbd (A[c]∗ ,B[c]∗ ). ∗ m,n ∗ ∗ Hence for every finite C0 ≤ C and for all m ≥ 1 the game Gbd (AC0 ,BC0 ) has ∗ ∗ a winning strategy, thus by Lemma 5.6 we have AC∗ ≡ BC∗ . ∗ ∗ Now let AC∗ ≡ BC∗ . By Lemma 5.6, this is equivalent to the property that ∗ m,s ∗ ∗ for every finite C0 ≤ C and for all m, s ≥ 1 the game Gbd (AC0 ,BC0 ) has a winning strategy. Fix m, s ≥ 1 and an element c ∈ Ss(C). By dualizing and m,s ∗ ∗ using a winning strategy for the game Gbd (A[c]∗ ,B[c]∗ ), player II can win the m,s game GIS (S(A)c,S(B)c). Hence for all m, s ≥ 1 and for every finite c ∈ Ss(C) m,s the game GIS (S(A)c,S(B)c) has a winning strategy, thus by Lemma 2.9 we have S(A)S(C) ≡ S(B)S(C). ¥ m,s Remark: The equivalence “all games GIS (A, B) have winning strategies if and m,s ∗ ∗ only if all games Gbd (A ,B ) have winning strategies” holds for arbitrary proabelian groups A, B; but only in the case of groups of finite exponent, the games m,s ∗ ∗ Gbd (A ,B ) are the appropriate ones to characterize elementary equivalence.

5.1.2 An independent set of invariants

In this subsection, we will prove Theorem 5.2. For this purpose, we will ﬁrst show that for an arbitrary proabelian group A , the LIS-theory of S(A) is determined ∗ ∗ by the family of the LG-theories (Th(A [n])n∈N of the n-torsion parts A [n] of A∗ (Proposition 5.11). Using this result we will prove Theorem 5.2, and we will see how to calculate the logical values of the invariants (up,n)p∈P,n∈N and

81 (Divp)p∈P occurring in Theorem 5.2 from A (but not yet in the easiest possible way). Remark: Let A be proabelian. If S(C) b S(A),S(B) is a common substructure, then the natural projections A → C → C/nC and B → C → C/nC factorize through nA respectively nB, thus S(C/nC) b S(A/nA),S(B/nB). If further [N] is an equivalence class of cardinality at most n in S(A), then the natural projection A → [N] factorizes through nA. Thus [N] lies in S(A/nA). This leads to the following observation: Observation 5.8. Let A be proabelian, let n ≥ 1 and let S(B) b S(A). Then for all sorts s ≤ n: ∼ Ss(A)Ss(B) = Ss(A/nA)Ss(B/nB).

By the above remark, the following lemma makes sense: Lemma 5.9. Let A, B be proabelian groups, let S(C) be a common substructure of S(A) and S(B). Then

S(A)S(C) ≡ S(B)S(C) ⇐⇒ for all n ≥ 1 : S(A/nA)S(C/nC) ≡ S(B/nB)S(C/nC).

In particular,

S(A) ≡ S(B) ⇐⇒ for all n ≥ 1 : S(A/nA) ≡ S(B/nB).

Proof: ⇐“ ”

For all n ≥ 1 : S(A/nA)S(C/nC) ≡ S(B/nB)S(C/nC)

⇐⇒ for all s, n ≥ 1 : Ss(A/nA)Ss(C/nC) ≡ Ss(B/nB)Ss(C/nC)

=⇒ for all n ≥ 1 : Sn(A/nA)Sn(C/nC) ≡ Sn(B/nB)Ss(C/nC) 5.8 ⇐⇒ for all n ≥ 1 : Sn(A)Sn(C) ≡ Sn(B)Sn(C)

⇐⇒ S(A)S(C) ≡ S(B)S(C).

To prove ⇒“, we point out that S(A/nA) is interpretable in S(A) : ” S(C/nC) S(C) For every open normal subgroup N of A, we have

[N] ⊆ S(A/nA) ⇐⇒ N ⊇ nA ⇐⇒ for all a ∈ [N]: ord(a) | n.

Thus Ss(A/nA) ⊆ Ss(A) is deﬁnable; the relations and constants are just the restrictions of those from Ss(A). ¥ Let again A, B be proabelian groups and let S(C) be a common substructure of S(A) and S(B). Then by Pontryagin duality, the dual (C/nC)∗ can be considered as a common subgroup of (A/nA)∗ and (B/nB)∗. Using the natural identiﬁcation from Corollary 1.15, this is the same as considering C∗[n] as a common subgroup of A∗[n] and B∗[n]. Together with Theorem 5.7 this leads to Lemma 5.10. Let A, B be proabelian groups, let S(C) be a common substructure of S(A) and S(B). Then for n ≥ 1

∗ ∗ S(A/nA)S(C/nC) ≡ S(A/nA)S(C/nC) ⇐⇒ A [n]C∗[n] ≡ B [n]C∗[n].

82 Proof:

5.7 ∗ ∗ S(A/nA)S(C/nC) ≡ S(A/nA)S(C/nC) ⇐⇒ (A/nA)(C/nC)∗ ≡ (B/nB)(C/nC)∗ 1.15 ∗ ∗ ⇐⇒ A [n]C∗[n] ≡ B [n]C∗[n]. ¥ Combining the last two lemmata, we get: Proposition 5.11. Let A, B be proabelian groups, let S(C) be a common substructure of S(A) and S(B). Then

∗ ∗ S(A)S(C) ≡ S(B)S(C) ⇐⇒ for all n ≥ 1 : A [n]C∗[n] ≡ B [n]C∗[n]. In particular S(A) ≡ S(B) if and only if for all n ≥ 1 holds A∗[n] ≡ B∗[n].

The proposition reduces the problem of ﬁnding elementary invariants for the LIS- theory of S(A) for a proabelian group A to the question of ﬁnding elementary ∗ ∗ invariants for the family of LG-theories of the m-torsion parts A [m] of A for all m ≥ 1. In the remainder of this section, we will indicate such elementary invariants and then deduce Theorem 5.2. Let A be an abelian torsion group, let as in Subsection 1.4.2

¡ n n+1 ¢ up,n(A) = dimFp (p A)[p]/(p A)[p] be the n-th Ulm invariant of A relative to p. For every m ≥ 1, the group A[m] decomposes as a direct sum of cyclic groups, in which up,n(A[m]) speciﬁes the number of summands isomorphic to Z/p n+1. Thus the entirety of all invariants up,n(A[m]) for n ∈ N, p ∈ P determines the m-torsion part A[m] up to isomorphism; it follows that the entirety of all logical values

W := (lv(up,n(A[m])))n∈N,p∈P,m≥1 determines1 all theories Th(A[m]). But the system W is not independent: Not all possible families of invariants occur as invariants of an abelian torsion group A. We will in the following extract an independent subset. Picture 5.2 on page 84 illustrates the situation. n+2 n Fix a prime p. As up,n(A) = up,n(A[p m]) for arbitrary m ≥ 1 and up,n(A[p m]) = 0 if p - m, we can replace W by the families (lv(u (A))) and (lv(u (A[pn+1]))) . p,n n∈N,p∈P | p,n {z n∈N,p∈P} =:V We take a closer look on the invariants in V. We have

n+1 ¡ n n+1 ¢ n up,n(A[p ]) = dimFp p (A[p ])[p] = dimFp ((p A)[p]) ;

n+1 thus the sequence (up,n(A[p ]))n∈N is monotonely decreasing. We already know the limit of the sequence from Deﬁnition 1.34:

n+1 lim up,n(A[p ]) = Divp(A), n→∞ where Divp(A) is one of the EF-invariants of A.

1 Note that there is an LG-sentences ϕ which is valid in A if and only if up,n(A[m]) = k.

83 A[p] =∼ (Z/p)(up,0(A[p]))

2 2 A[p2] =∼ (Z/p)(up,0(A[p ])) × (Z/p2)(up,1(A[p ]))

3 3 3 A[p3] =∼ (Z/p)(up,0(A[p ])) × (Z/p2)(up,1(A[p ])) × (Z/p3)(up,2(A[p ])) 84

n n n n A[pn] =∼ (Z/p)(up,0(A[p ])) × (Z/p2)(up,1(A[p ])) × (Z/p3)(up,2(A[p ])) × ... × (Z/pn)(up,n(A[p ]))

(u (A)) 2 (u (A)) 3 (u (A)) n (u (A)) (Divp(A)) A ≡ (Z/p) p,0 × (Z/p ) p,1 × (Z/p ) p,2 × ... × (Z/p ) p,n × ... × Zp

Picture 5.2 Further we have

n+1 up,n(A[p ]) = dim ((pnA)[p]) Fp ¡ ¢ ¡ ¢ (∗) n n+1 n+1 = dimFp (p A)[p]/(p A)[p] + dimFp (p A)[p] n+2 = up,n(A) + up,n+1(A[p ])

Now we distinguish two cases:

Case 1: The N-sequence ((up,n(A))n∈N, Divp(A)) is infinite. If the sequence (up,n(A))n∈N does not eventually become zero, then necessarily n+1 up,n(A[p ]) ≥ ℵ0 for all n ∈ N because of (∗), and so as well Divp(A) ≥ ℵ0. In this case the logical values (lv(un,p(A)))n∈N already determine all values attached to the prime p in V. Case 2: The N-sequence ((up,n(A))n∈N, Divp(A)) is finite. If the sequence (up,n(A))n∈N eventually becomes zero, the value Divp(A) = n+1 limn→∞ up,n(A[p ]) may be finite. In this case, using (∗), one can calculate all the logical values in V attached to the prime p from (lv(up,n(A)))n∈N, Divp(A)). Definition 5.12. For an abelian torsion group A, define ³¡ ¢ ¡ ¢´ Up(A) = lv(up,n(A)) n∈N, lv Divp(A) .

We have just seen that we can replace the original set of invariants W from page 83 by the family of N-sequences

(Up(A))p∈P.

But this system of invariants is still not independent because of the dependence of lv(Divp(A)) from the sequence (lv(un,p(A)))n∈N in Case 1 above. We deﬁne:

Definition 5.13. An N-sequence (lv(un))n∈N is adequate if (un)n∈N is finite, or if it is infinite and lv(u∞) = ∞.

Remark:

• We have just seen that for an abelian torsion group A, all N-sequences Up(A) are adequate. • Every equivalence class of N-sequences of logical values of cardinals contains exactly one adequate representative. Our considerations show that two abelian torsion groups A, B have equivalent N-sequences Up(A), Up(B) for every prime p if and only if Th(A[m]) = Th(B[m]) for every m ∈ N. It remains to show that given arbitrary families (Up)p∈P of N-sequences, there is an abelian torsion group A with U p(A) = U p for p ∈ P; by the above, Up(A) has to be the adequate representative of U p. But given any family of N-sequences (UpL)p∈P with Up = ((lv(up,n))n∈N, Divp), one can easily check that the group A = p∈P Ap with M n+1 (up,n) (Divp) Ap = (Z/p ) ⊕ (µp∞ ) n∈N

85 does the job, i.e. U p = U p(A) holds for every prime p. Noticing that the dual of a direct sum of discrete abelian torsion groups is the ∗ direct product of the dual groups, and that (µp∞ ) = Zp, we ﬁnally get an (improved) version of Theorem 5.2: Theorem 5.2. (Second version) Let A be proabelian. Then S(A) is elementarily equivalent to S(B) for the direct product

Y ∗ Y ∗ n+1 up,n(A ) Divp(A ) B = (Z/p ) × Zp . n∈N, p prime p prime

The map ∗ A → (Up(A ))p∈P induces a bijection between theories of complete inverse systems of proabelian groups and families (U p)p∈P of equivalence classes of N-sequences Up.

Proof: By the above,

S(A) ≡ S(B) ⇐⇒5.11 for all m ∈ N : A∗[m] ≡ B∗[m] ∗ ∗ ⇐⇒ for all p ∈ P : U p(A ) = U p(B ).

The group Y ∗ Y ∗ n+1 up,n(A ) Divp(A ) B = (Z/p ) × Zp n∈N, p prime p prime

∗ ∗ satisﬁes U p(B ) = U p(A ), thus S(A) ≡ S(B). ¥

Remark 5.14. Let A, B be proabelian groups, for p ∈ P let Ap,Bp be the p- Sylow groups of A, B. From the above follows

S(A) ≡ S(B) ⇐⇒ for all p ∈ P : S(Ap) ≡ S(Bp).

We ﬁnish this section with an example that we already announced before:

Example 5.15. Let D be the discrete abelian torsion group with generators x, y1, 2 ∗ y2,... subject to relations 0 = px, x = py1 = p y2 = .... Let D := D be the Q n+1 Pontryagin dual of D, and let P := n∈N(Z/p ). Then (cf. Example 5.4) 1) D does not decompose as a direct product of procyclic groups. 2) S(D) ≡ S(P )

Proof: It is well known that the Ulm invariants of D are up,α(D) = 1 for 0 ≤ α ≤ ω and up,α(D) = 0 for α > ω, see for example [Ka], page 31. In every direct sum A of cyclic groups and Prufer¨ groups we have uα,p(A) = 0 for α ≥ ω; thus D can’t be a direct sum of cyclic groups and groups. Hence D does not decompose as a direct product of procyclic groups. But for both D and ∗ L n+1 ∗ P = n∈N Z/p we have lv(up,n(D)) = lv(up,n(P )) = 1 for n ∈ N; thus ∗ U p(D) = U p(P ), and hence S(D) and S(P ) are elementarily equivalent. ¥

86 Remark 5.16. It is well known (cf. Exercise 42 on page 33 in [Fu]) that if (un)n∈N is an arbitrary sequence of cardinals ≤ ℵ0 from which inﬁnitely many are not equal to 0, then there is a countable abelian p-group A such that up,n(A) = up,n for n ∈ N and uω,p(A) = 1. As above, A does not decompose as a direct sum of cyclic groups and Prufer¨ groups. For the direct sum M T := (Z/p n+1)un n∈N

∗ ∗ we have U p(T) = U p(A). Thus the complete inverse systems S(A ) and S(T ), which are both countable, are elementarily equivalent, but not isomorphic.

5.2 Connecting A, A∗ and S(A) for a proabelian group A

Recall that the elementary theory of an arbitrary abelian group is determined by its elementary EF-invariants, cf. Fact 1.35. In Subsection 5.2.1, we will show that for abelian torsion groups A, a proper subset of the elementary EF-invariants is suﬃcient to determine the elementary theory of A, and that this subset is already determined by the theories Th(A[m]) of the m-torsion parts of A. Thus it follows from Proposition 5.11 that two complete inverse systems S(A),S(B) of proabelian groups A, B are elementarily equivalent if and only if the dual groups A∗,B∗ of A, B are elementarily equivalent. In Subsection 5.2.2, we will ∗ ∗ use Theorem 1.41 to show that the N-sequence Up(A ) of A consists of a subset of the EF-invariants of A.

5.2.1 Equivalence of abelian torsion groups

We need the following lemma: Lemma 5.17 (Lemma 1.6 in [EkF]). 1) Let A be an abelian torsion group. Then for all primes p and all k ≥ 0:

k k+1 p A = p A ⇐⇒ for all n ≥ k : up,n(A) = 0.

k k+1 k k+1 2) If p A 6= p A for all k ≥ 0, then dimFp (p A/p A) is inﬁnite for all k ≥ 0. Corollary 5.18. Let A be an abelian torsion group and let p be a prime. Further n n+1 let Tfp(A) = lim dimFp (p A/p A) be as in Deﬁnition 1.34. Then n→∞   0 if (up,n(A))n∈N becomes eventually zero lv(Tf (A)) = p  ∞ otherwise.

In particular, if A, B are abelian torsion groups with A[m] ≡ B[m] for all m ≥ 1, then lv(Tfp(A)) = lv(Tfp(B)).

Proof: By the above lemma, Tfp(A) is either inﬁnite or zero, and Tfp(A) = 0 if and only if (up,n(A))n∈N eventually becomes zero. Thus lv(Tfp(A)) is already

87 determined by the sequence (lv(up,n(A)))n∈N of A. If A, B are abelian torsion groups with A[m] ≡ B[m] for all m ≥ 1, then U p(A) = U p(B), and thus the sequences (lv(up,n(A)))n∈N, (lv(up,n(B)))n∈N are the same. ¥ Now we look at the remaining EF-invariant Exp(A). Obviously Exp(A) = 0 if and only if almost all Ap = 0 and if those Ap with Ap 6= 0 have ﬁnite exponent.

Lemma 5.19. Let A be an abelian torsion group. For every prime p let Ap be the p-part of A. Then

1) Ap = 0 ⇐⇒ the N-sequence Up(A) is everywhere zero, and 2) Ap has ﬁnite exponent ⇐⇒ Divp(A) = 0.

n Proof: For 1), assume that Ap 6= {0}; let for instance x ∈ Ap with ord(x) = p . Consider y = pn−1x ∈ A[p]. If y has finite p-height, say the p-height is pk, then k k+1 y ∈ p A[p], but y 6∈ p A[p], so uk,p(A) 6= 0. If y has infinite p-height, then k y ∈ p A[p] for all k ∈ N, and thus Divp(A) ≥ 1. The other direction is trivial. For 2), assume that Divp(A) 6= 0. This implies that there are arbitrary large n n ∈ N with (p Ap)[p] 6= 0. Thus there are arbitrary large n ∈ N such that n Ap[p ] 6= 0, hence Ap has infinite exponent. For the other direction, assume that i Ap has infinite exponent. Let (ai)i∈N be a sequence in Ap with ord(ai) ≥ p . We i i−1 i−1 can assume that ord(ai) = p . Then p ai ∈ p A[p], and hence Divp(A) = n lim dimF (p A[p]) ≥ 1. ¥ n→∞ p From the lemma follows immediately: Corollary 5.20. Let A be an abelian torsion group. Then Exp(A) = 0 if and only if all but finitely many of the Ulm invariants up,n(A) for n ∈ N, p ∈ P are zero and if Divp(A) = 0 for all primes p. In particular, if A, B are abelian torsion groups with A[m] ≡ B[m] for all m ≥ 1, then Exp(A) = Exp(B).

From this we obtain: Proposition 5.21. Let A, B be abelian torsion groups. Then

A ≡ B ⇐⇒ for all m ∈ N : A[m] ≡ B[m].

Proof: “⇒” is clear. For the other direction, assume that A[m] ≡ B[m] holds for all m ∈ N. Then U p(A) = U p(B) and thus by the last two corollaries, A and B have the same EF-invariants. Hence by Fact 1.35, the groups A and B are elementarily equivalent. ¥ Remark: Note that the statement is obviously false for arbitrary abelian groups: For example, we have Q[pn] =∼ 0[pn] for all n ∈ N, but Q 6≡ 0. Together with Corollary 5.11, the lemma proves: Theorem 5.22. Let A, B be proabelian groups. Then

S(A) ≡ S(B) ⇐⇒ A∗ ≡ B∗.

Elementary substructures

Lemma 5.23. Let A ⊆ B be abelian torsion groups. Then A is pure in B if and only if for all n ∈ N, the n-torsion part A[n] is pure in B[n].

88 Proof: One direction is clear: If A[n] v B[n] for all n ∈ N, then obviously A v B. For the other direction, notice that for two torsion groups holds A v B if and only if for the p-parts holds Ap v Bp for every prime p. Hence we can restrict on p-groups and p-purity. Thus let A, B be p-groups, let a ∈ A[pn], b ∈ B[pn] and let k ∈ N such that pkb = a. If ord(b) = pk, then a = 0 and there is nothing to show. If ord(b) > pk, then choose a¯ ∈ A with pka¯ = a; then ord(¯a) = ord(b) < pn, and hence a¯ ∈ A[pn]. If ﬁnally ord(b) < pk, then (pk mod ord(b)) · b = a. Thus there is a¯ ∈ A with (pk mod ord(b)) · a¯ = a and ord(¯a) = ord(b) ≤ pn. Now a¯ does also pk-divide a. ¥ From this together with Theorem 1.35 and Proposition 5.21 we get Lemma 5.24. Let A ⊆ B be abelian torsion groups. Then A ≺ B ⇐⇒ ∀m ∈ N : A[n] ≺ B[n].

Combined with Proposition 5.11 (letting S(C) = S(A)), this leads to Theorem 5.25. Let S(A) b S(B) be complete inverse systems of proabelian groups. Then S(A) ≺ S(B) ⇐⇒ A∗ ≺ B∗.

5.2.2 Dualizing elementary invariants

First we show that if A, B are proabelian and A∗,B∗ are their Pontryagin duals, then A ≡ B ⇐⇒ A∗ ≡ B∗. In fact, after what we learned in Subsection 1.4.2, ∗ the result is immediate. We use this to show that the N-sequences U p(A ) consist of some of the elementary EF-invariants of A. Let as in Subsection 1.4.2 be A+ = Hom(A, Q/Z) the character group of an arbitrary abelian group A and Th(A)∗ be the dual theory to Th(A). If A happens to be proﬁnite or torsion, we denote by A∗ as usual the Pontryagin dual2 of A. Theorem 5.26. Let A, B be proabelian groups. Then A∗ ≡ B∗ ⇐⇒ A ≡ B

Proof: As A∗ is an abelian torsion group, we have (A∗)+ = (A∗)∗ =∼ A; accordingly for B. Thus A ≡ B ⇐⇒ Th(A) = Th(B) ⇐⇒ Th((A∗)+) = Th((B∗)+) ⇐⇒1.41 Th((A∗)+)∗ = Th((B∗)+)∗ ⇐⇒1.41 Th(A∗) = Th(B∗) ⇐⇒ A∗ ≡ B∗. ¥ ∗ Next we will show how to calculate the invariants U p(A ) directly from A. 2Note that for an proabelian group A, the groups A∗ and A+ do in general not coincide: In A∗, all homomorphisms are continuous, while in A+ arbitrary homomorphisms are allowed. In contrast, if A is an abelian torsion group, then A∗ and A+ coincide.

89 Deﬁnition 5.27. Let A be a proabelian group. Let for a prime p

Vp(A) = ((up,n(A))n∈N, Tfp(A)),

n n+1 where Tfp(A) = lim dimFp (p A/p A) and the up,n(A) are as in Deﬁnition n→∞ 1.34. We call the elements of the N-sequences Vp(A) the proﬁnite invariants of A.

Now we are able to give a ﬁnal version of theorem 5.2: Theorem 5.2. (Third version) Let A be proabelian. Then S(A) is elementarily equivalent to S(B) for the direct product

Y Y Tf (A) n+1 up,n(A) p B := (Z/p ) × Zp n∈N, p prime p prime

The map A → (Vp(A))p∈P induces a bijection between theories of complete inverse systems of proabelian groups and families (Vp)p∈P of equivalence classes of N-sequences.

Proof: By Theorem 5.22 and Theorem 5.26, two complete inverse systems S(A) and S(B) are LIS-equivalent if and only if the proabelian groups A and B are LG-equivalent. From this and from the last version of the above theorem on page 86 we know that A is elementarily equivalent to the direct product

Y ∗ Y ∗ n+1 up,n(A ) Divp(A ) B := (Z/p ) ⊕ Zp . n∈N, p prime p prime

By Fact 1.36, A is also elementarily equivalent to the direct sum

M ∗ M ∗ n+1 (up,n(A )) (Divp(A )) C := (Z/p ) ⊕ Zp . n∈N, p prime p prime

It follows that A has the same elementary EF-invariants as C. Using Lemma 1.38 one easily sees

∗ lv(up,n(A)) = lv(up,n(C)) = lv(up,n(A )) and ∗ lv(Tfp(A)) = lv(Tfp(C)) = lv(Divp(A )).

¥ To conclude this section, we show what the EF-invariants of the dual theory T∗ of a complete theory T of abelian groups are. Note that also in [PRZ], there is a procedure how to deduce T∗ from T: Given T, one can determine a set of so-called index formulas axiomatizing T; these index formulas can be dualized, and the duals are then an axiomatization of T∗. We use an algebraic approach starting from the more common EF-invariants of T resulting in the EF-invariants of T∗. We need the following fact about homomorphism groups:

90 Fact 5.28 (Theorem 43.1 in [Fu]). If (Ai)i∈I is a family of abelian groups and C is an abelian group, then M Y ∼ Hom( Ai, C) = Hom(Ai, C). i∈I i∈I Proposition 5.29. Let A be an abelian group. Then the EF-invariants of A+ are the dual invariants of the EF- invariants of A, i.e.

+ lv(up,n(A )) = lv(up,n(A)) + lv(Divp(A )) = lv(Tfp(A)) + lv(Tfp(A )) = lv(Divp(A)) Exp(A+) = Exp(A) for n ∈ N and p ∈ P.

Proof: We know from Theorem 1.37 that A is elementarily equivalent to the direct sum M M M n+1 (up,n(A)) (Tfp(A)) (Divp(A)) δ B = (Z/p ) ⊕ Zp ⊕ µp∞ ⊕ Q , n∈N, p prime p prime p prime where ½ 0 if Exp(A) = 0 δ = 1 if Exp(A) = ∞ . We want to calculate the EF-invariants of B+. We already know that (Z/p n+1)+ =∼ n+1 + ∗ ∼ Z/p and that µp∞ = µp∞ = Zp. By Theorem 1.42, + ∗ + + + Zp = (µp∞ ) = (µp∞ ) Â µp∞ , + thus Zp is elementarily equivalent to µp∞ . The group Hom(Q, Q/Z) is obviously divisible because Q/Z is. Further it is torsion free because Q is divisible: Let 0 6= f ∈ Hom(Q, Q/Z), say f(q) = q r + Z 6= 0. Then nf( n + Z) = r + Z 6= 0, hence nf 6= 0. Further Hom(Q, Q/Z) is non-trivial. Thus Hom(Q, Q/Z) has the same invariants as Q and is thus elementarily equivalent to Q. Hence by Fact 5.28 and Fact 1.36, we know that B+ and thus also A+ is elementarily equivalent to M M M n+1 (up,n(A)) (Divp(A)) (Tfp(A)) δ C = (Z/p ) ⊕ Zp ⊕ µp∞ ⊕ Q . n∈N, p prime p prime p prime Using Lemma 1.38, one easily checks that the EF-invariants of C are as desired. ¥

5.3 Some model theory of proabelian groups

5.3.1 Categoricity

Observation 5.30.Q Let p be a prime and (un)n∈N be a sequence of cardinal n+1 un ∼ numbers, let A = n∈N(Z/p ) . Then A =6 A × Zp.

91 Proof: If the sequence (un)n∈N becomes eventually zero, the groups are not even elementarily equivalent. In general, note that in A × Zp there exists an inﬁnite sequence U0 ≥ U1 ≥ ... of open normal subgroups such that (A × ∼ n+1 Zp)/Un = Z/p , while no such sequence exists in A (cf. [C1], last theorem in the appendix). ¥

Lemma 5.31. Let p be a prime, let A be a proabelian group and Ap the p-Sylow subgroup of A. Then

Vp(A) is ﬁnite ⇐⇒ Tor(Ap) has ﬁnite exponent.

Proof: As Vp(A) = Vp(Ap), we can assume that A is a pro-p-group. “⇐” Assume that the torsion part Tor(A) of A has ﬁnite exponent, for instance Tor(A) = A[pn]. Then Tor(A) is a closed subgroup of A and A/Tor(A) is torsion ∼ κ free, hence for some cardinal κ we have A/Tor(A) = Zp , which is a free abelian pro-p-group. Thus the sequence

κ 0 → Tor(A) → A → Zp → 0

∼ k splits and A = Tor(A) × ZpQ. As Tor(A) has ﬁnite exponent, it decomposes as ∼ n+1 un a direct product Tor(A) = n∈N(Z/p ) , in which almost all un are zero. But the Ulm invariants of A equal those of Tor(A), i.e. un = up,n(A), and thus Vp(A) is ﬁnite.

“⇒”Let Vp(A) be ﬁnite. By the preceding theorems, A is elementarily equivalent to the direct product Y n+1 up,n(A) Tfp(A) B = (Z/p ) × Zp . n∈N

As Vp(A) is ﬁnite, there is k ∈ N suchQ that up,n(A) = 0 for n ≥ k; thus for the n+1 up,n(A) k torsion part of B we have Tor(B) = n∈N(Z/p ) and Tor(B) = B[p ]. k k+l Consider the formula ϕl = ∀x(p x 6= 0 → p x 6= 0). Then B ² ϕl for l ≥ 1. Thus A ² ϕl for l ≥ 1, which implies that every torsion element in A has order at most pk. Thus Tor(A) has ﬁnite exponent. ¥

Theorem 5.32. Let A be proabelian. Then the theory TA of S(A) is ℵ0- categorical if and only if all N-sequences Vp(A) are ﬁnite, if and only if for all p-Sylow subgroups Ap of A the torsion part Tor(Ap) has ﬁnite exponent.

Proof: The second equivalence follows directly from Lemma 5.31. Two proabelian groups are isomorphic if and only if for all primes p their p-Sylow groups are isomorphic. Hence we can restrict on pro-p-groups.

“⇐” Let A be a pro-p-group, and let Vp(A) be finite. Then by the last lemma, for every model S(B) of TA, the torsion part Tor(B) has finite exponent; thus ∼ Qas in the last lemma, the torsion part splits as a direct factor. Hence B = n+1 un u∞ n∈N(Z/p ) × Zp for some N-sequence (un)n∈N with almost all entries being zero. Thus for every model S(B) of TA, the group B is a direct product of procyclic groups. Moreover lv(un) = lv(up,n(A)) for n ∈ N and lv(u∞) = lv(Tfp(A)). Hence if S(B) is countable, all profinite invariants are countable and thus uniquely determined by their logical value.

92 Q ∼ n+1 up,n ∼ “Q⇒” Assume that Vp(A) is infinite. Let B1 = n∈N(Z/p ) and B2 = n+1 up,n n∈N(Z/p ) ×Zp, where all up,n are countable and lv(up,n) = lv(up,n(A)) for all n ∈ N. Then both S(B1) and S(B2) are countable, and by Theorem 5.2, they are elementarily equivalent. But by Observation 5.30, they are not isomorphic. ¥ Remark: • The proof shows that if the torsion part of A does not have finite exponent, then there are even two direct products of procyclic groups whose complete inverse systems are elementarily equivalent to the one of A, but not isomorphic. Further, Remark 5.16 shows that there is also a model for TA that does not decompose as a direct product of procyclic groups, cf. with the proof of the next theorem. • Alternatively, to prove the non-categoricity, one could use the Ryll-Nardzewski Theorem: Infinitely many of the formulas ϕn(x) stating

[x] =∼ Z/pZ ∧ ∃y ([y] =∼ Z/pnZ ∧ y C³ x)

for n ∈ N are modulo TA pairwise not equivalent.

Theorem 5.33. Let A be proabelian. Then the theory TA of S(A) is κ-categorical for some κ ≥ ℵ1 if and only if all N-sequences Vp(A) are finite, and at most one of the profinite invariants of A is infinite.

Proof: First assume that at least two of the proﬁnite invariants of A are in-

ﬁnite; for example, let up1,n1 (A) and up2,n2 (A) be inﬁnite. Choose direct prod-

ucts B1 and B2 of procyclic groups with up1,n1 (B1) = κ, up2,n2 (B1) = ℵ0,

up1,n1 (B2) = ℵ0 and up2,n2 (B2) = κ such that moreover S(A) ≡ S(B1) ≡ S(B2); then S(B1),S(B2) are not isomorphic, and both S(B1) and S(B2) have cardinality κ.

Now assume that for some prime p the N-sequence Vp(A) is inﬁnite. Using Remark 5.16, we see that there is a countable model S(B) of TA such that ∗ up,ω(B ) = 1 , thus B does not decompose as a direct product of procyclic κ ∗ groups. Then for C = B × Zp we have as well up,ω(C ) = 1, thus C does not decompose as a direct product of procyclic groups either. Further |S(C)| = κ and S(C) ² TA. But there is also a model S(D) of TA of cardinality κ such that D does decompose as a direct product of procyclic groups, thus TA is not κ-categorical.

Finally, assume that all N-sequences Vp(A) are ﬁnite and that only one of the

profinite invariants of A is infinite; assume for example that lv(up0,n0 (A)) = ∞, and that all other invariants are finite. As in the last theorem, we see that for every prime p, the p-Sylow group Bp of any B with S(B) ≡ S(A) decomposes as Q Tf (B) ∼ n+1 up,n(B) p a direct product Bp = n∈N(Z/p ) ×Zp with almost all exponents being zero. As B ≡ A, we have lv(up,n(B)) = lv(up,n(A)) for all n ∈ N and

lv(Tfp(B)) = lv(Tfp(A)). If S(B) has cardinality κ, then necessarily un0,p0 (B) = κ, and all other exponents are ﬁnite and thus equal to the proﬁnite invariants of A. This shows that there is up to isomorphism only one model of cardinality κ. ¥

93 5.3.2 Saturated models and stability

In the whole subsection, let κ be a cardinal ≥ ℵ1.

Deﬁnition 5.34. Let A be an abelian group, let p((xi)i∈I ) be a (possibly incom- plete) type over C ⊆ Tor(A). We call p a torsion type if and only if for every . i ∈ I there is ni ≥ 1 such that p ` nixi = 0. Deﬁnition 5.35. Let A be an abelian group. We say that A is κ-torsion-saturated if A is a torsion group and all torsion types p(x) over parameter sets C ⊆ A of cardinality less than κ are realized in A. Observation 5.36. 1) If A is κ-torsion-saturated and p is a torsion type over a parameter set C of cardinality less than κ whose variables are contained in {xi | i < κ} , then p is realized in A. 2) As κ is uncountable, then for A to be κ-torsion-saturated it is equivalent that all torsion types p(x) over subgroups C of A (equivalently: over elementary substructures C ≺ A) of cardinality less than κ are realized in A.

Theorem 5.37. Let A be a proabelian group and κ ≥ ℵ1. Then S(A) is κ- saturated if and only if A∗ is κ-torsion-saturated.

Proof: “⇒” Let S(A) be κ-saturated. We have to show that A∗ is κ-torsion- saturated. Let C∗ ≤ A∗ be a subgroup of cardinality < κ, let p(x) be a torsion type over C. Realize p via b∗ in some B∗ Â A∗. As p is torsion, b∗ ∈ Tor(B∗), and by Corollary 1.40, we have A∗ ≺ Tor(B∗) ≺ B∗; thus we can assume that B∗ is torsion. ∗ ∗ ∗ ∗ ∗ Let hb i = {b1 = b , . . . , bn} be the subgroup generated by b , let q(x1, . . . , xn) ∗ ∗ ∗ ∗ = tp(b1, . . . , bn/C ). We are done if we can show that q is realized in A . ∗∗ ∗∗ ∗∗ ∗ ∗ Identify A with A , and let B := B ,C := C , [b] = hb i = {b1, . . . , bn}; then by Pontryagin duality and Theorem 5.25, we have S(C) b S(A) ≺ S(B), and [b] is a full equivalence class of S(B). Now realize tp(b1, . . . , bn/S(C)) in S(A) via a1, . . . , an; then the map f : S(C)∪{b1, . . . , bn} → S(C)∪{a1, . . . , an}, which is the identity on S(C) and maps bi to ai for 1 ≤ i ≤ n, is a partial elementary map from S(B) to S(B). Choose a strongly κ-homogeneous S(D) Â S(B); then f can be extended to an automorphism, which we denote again by f, of S(D). By Pontryagin duality, the map G(f)∗ : D∗ → D∗, which arises from f by applying the composed functor ∗ ◦ G, is an automorphism as well and by Theorem 5.25, ∗ ∗ ∗ ∗ ∗ ∗ we further have D Â B . Thus the image a1, . . . , an of b1, . . . , bn is the desired realization of q in A∗. “⇐” Let A∗ be κ-torsion-saturated. We have to show that S(A) is κ-saturated. Let p(x) be a type over a parameter set S0 ⊆ S(A) of cardinality < κ, where x is a variable of an arbitrary sort. As κ is uncountable, we can assume that S0 = S(C) is a substructure of S(A). Realize p via b in S(B) Â S(A); let [b] = {b1, . . . , bn}. It suﬃces to show that q(x1, . . . , xn) := tp(b1, . . . , bn) can be realized in S(A) (we choose the xi of the same sort as x). ∗ ∗ ∗ ∗ ∗ Dualizing and Theorem 5.25 yields C ≤ A ≺ B , and further [b] = {b1, ∗ ∗ ∗ ∗ . . . , bn} is a ﬁnite subgroup of B . As B is a torsion group, the type tp(b1,..., ∗ ∗ ∗ ∗ ∗ bn/C ) is a torsion type and is thus realized by some tuple a1, . . . , an in A . ∗ ∗ ∗ ∗ ∗ ∗ ∗ The map f : C ∪ {b1, . . . , bn} → C ∪ {a1, . . . , an}, which is the identity on

94 ∗ ∗ ∗ ∗ C and maps bi to ai for 1 ≤ i ≤ n, is a partial elementary map from B to B∗. Choose a strongly κ-homogeneous D∗ Â B∗; then f ∗ can be extended to an automorphism, again denoted by f ∗, of D∗. Again by Corollary 1.40, the torsion part Tor(D∗) is an elementary extension of B∗. Further f ∗ restricts to an automorphism of Tor(D∗). Hence we can assume that D∗ is a torsion group. By Pontryagin duality, the map S(f ∗∗): S(D) → S(D), which arises from f by applying the composed functor S ◦∗, is an automorphism, and by Theorem 5.25, we have S(D) Â S(B). Thus the image a1, . . . , an of b1, . . . , bn is the desired realization of q in S(A). ¥

Theorem 5.38. If A is a proabelian group, then TA := Th(S(A)) is stable.

Proof: Let M be a subset of cardinality κ of a sufficiently saturated model S(B) ℵ0 of TA, where κ is a cardinal with κ = κ. We can assume that M = S(C) is a substructure of S(B). Let n be an arbitrary sort. We have to show that the elements of sort n of S(B) realize at most κ different types over S(C). As κ is uncountable, it is equivalent to show that there are at most κ types of tuples (a1, . . . , an) in S(B) over S(C), where the ai are pairwise different elements of sort n and form an equivalence class in S(B). Let (a1, . . . , an) and (b1, . . . bn) be n-tuples of sort n such that [a1] = {a1, . . . , an} and [b1] = {b1, . . . , bn} are full equivalence classes. As in the previous proof, one sees that if the map which is the identity on S(C) and maps ai to bi for 1 ≤ i ≤ n is a partial elementary map from S(B) to S(B), then there is a map between ∗ ∗ ∗ the dual groups [a1] and [b1] which is a partial elementary map over C . ∗ ∗ ∗ ∗ ∗ ∗ ∗ Analogously, if {a1, . . . , an} ≤ B and {b1, . . . bn} ≤ B are subgroups of B ∗ ∗ ∗ such that the map which is the identity on C and maps ai to bi for 1 ≤ i ≤ n is a partial elementary map from B∗ to B∗, then there is a map between the ∗ ∗ ∗ ∗ ∗ ∗ equivalence classes {a1, . . . , an} and {b1, . . . bn} which is a partial elementary map of S(B) over S(C). The group B∗ is κ-stable; hence there are at most κ different n-types over C∗. It follows that there are at most n!·κ = κ types of n-tuples a1, . . . an with pairwise different ai in S(B). ¥

95 Bibliography

[C1] Z. Chatzidakis, Model theory of proﬁnite groups, Dissertation, Yale University, 1984 [C2] Z. Chatzidakis, Model theory of groups having the Iwasawa property, Illinois J. Math. Volume 42, Issue 1 (1998), 70-96

[C3] Z. Chatzidakis, Properties of forking in ω-free Pseudo-Algebraically closed fields, J. Symbolic Logic Volume 67, Issue 3 (2002), 957-996 [CDM] G. Cherlin, L. van den Dries, and A. Macintyre, The elementary theory of regularly closed fields, preprint, 1980 [D] S. Demushkin, On the maximal p-extension of a local field (Russian), Izv. Akad. Nauk, USSR. Math. Ser., 25 (1961), 329-346 [Ef] I. Efrat, Demuˇskin fields with valuations, Math. Z. 243 (2003), 333-353

[Ef2] I. Efrat, Finitely generated pro-p Galois groups of p-Henselian ﬁelds, J. Pure and Applied Algebra, Volume 138, Issue 3 (1999), 215-228 [Ef3] I. Efrat, Small maximal pro-p Galois groups, Manusc. Math. 95 (1998), 237-249

[Ef4] I. Efrat, Valuations, Orderings, and Milnor K-Theory, Mathematical Surveys and Monographs 124, American Mathematical Society, Prov- idence, RI, 2006 [EkF] P. C. Eklof, E. R. Fisher, The elementary theory of abelian groups, Ann. Math. Logic 4 (1972), 115-171 [EnP] A. J. Engler, A. Prestel, Valued Fields, Springer Monographs in Math- ematics, Springer-Verlag, Berlin, 2005 [Fu] L. Fuchs, Inﬁnite Abelian Groups, Volume I, Pure and Applied Math- ematics, 36. New York-London: Academic Press. XI, 290 p. (1970)

[GSZ] P. Gille, T. Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, 101, Cambridge University Press, Cambridge, 2006

[Hz] I. Herzog, Elementary duality of modules, Trans. Amer. Math. Soc. 340 (1993), no. 1, 37-69

96 [JL] M. Jarden and A. Lubotzky, Elementary equivalence of profinite groups, Bull. London Math. Soc. 40 (2008), 887-896 [JP] C.U. Jensen, A. Prestel, Realization of finitely generated profinite groups by maximal abelian extensions of fields, J. Reine Angew. Math. 447 (1994), 201-218

[K1] J. Koenigsmann, A Galois code for ﬁelds, Preprint, 2010 [K2] J. Koenigsmann, Relatively projective groups as absolute Galois groups, Israel J. Math. 127 (2002), 93-129 [K3] J. Koenigsmann, From p-rigid elements to valuations (with a Galois- characterization of p-adic ﬁelds), J. Reine Angew. Math. 465 (1995), 165-182

[K4] J. Koenigsmann, Pro-p Galois groups of rank ≤ 4, Manusc. Math. 95 (1998), no. 2, 251-271

[K5] J. Koenigsmann, Projective extensions of ﬁelds, J. London Math. Soc. (2) 73 (2006), 639-656 [K6] J. Koenigsmann, Solvable absolute Galois groups are metabelian, In- vent. Math. 144 (2001), no. 1, 1-22

[K7] J. Koenigsmann, Products of absolute Galois groups, Int. Math. Res. Not. 2005, no. 24, 1465-1486 [K8] J. Koenigsmann, Elementary Characterization of Fields by their Ab- solute Galois Groups, Sibirian Advances in Mathematics (2004), v. 14, N3, 1-26 [Ka] I. Kaplansky, Inﬁnite abelian groups, University of Michigan press, revised edition, 1969

[Ke] I. Kersten, Brauergruppen von K¨orpern, Vieweg + Teubner, 1990 [KK] G. Kreisel, J.-L. Krivine, Elements de logique mathematique, Durrod, Paris, 1967

[KPR] V. Kuhlmann, Matthias Pank, Peter Roquette, Immediate and purely wild extensions of valued ﬁelds, Manusc. Math. 55 (1986), 39-67

[La] S. Lang, Algebra, Addison-Wesley Publishing Company, Inc., 1993 [Lab1] J. P. Labute, Classiﬁcation of Demu˘skin Groups, Can. J. of Math. 19 (1967), 106-132

[Lab2] J. P. Labute, Demu˘skin Groups of rank ℵ0, Bull. Soc. Math. France 94 (1966), 211-244 [Lam] T. Lam, Introduction to quadratic forms over ﬁelds, Graduate Studies in Mathematics, 67. American Mathematical Society, Providence, RI, 2005

97 [MW] J. Miná˘c,Roger Ware, Demu˘skin Groups of Rank ℵ0 as absolute Galois Groups, Manusc. Math. 73 (1991), 411-421 [NS] N. Nikolov, D. Segal, On finitely generated profinite groups I: strong completeness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171-238

[NSW] J. Neukirch, Alexander Schmidt, Kay Winberg, Cohomology of number ﬁelds, Second edition. Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences], 323. Springer-Verlag, Berlin, 2008

[P1] M. Prest, Duality and pure-semisimple rings, J. London Math. Soc. (2) 38 (1988), no. 3, 403-409 [P2] M. Prest, Remarks on elementary duality, Annals of Pure and Applied Logic 62 (1993), 183-205

[Pr] A. Prestel, Einfuhrung¨ in die mathematische Logik und Modelltheo- rie, (German) [Set theory for the mathematician], Vieweg Studium: Grundkurs Mathematik [Vieweg Studies: Basic Mathematics Course], 58. Friedr. Vieweg & Sohn, Braunschweig, 1985

[PRZ] M. Prest, P. Rothmaler, M. Ziegler, Extensions of elementary duality, J. Pure Appl. Algebra 93 (1994), no. 1, 33-56 [Q] F. Quigley, Maximal subﬁelds of an algebraically closed ﬁeld not containing a given element, Proc. Amer. Math. Soc. 13 (1962), 562-566

[Ro] D. J. S. Robinson, A course in the theory of groups, Second edition. Graduate Texts in Mathematics, 80. Springer-Verlag, New York, 1996 [RZ] L. Ribes, P. Zalesskii, Proﬁnite groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathemat- ics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 40. Springer-Verlag, Berlin, 2000 [Se] J.-P. Serre, Galois Cohomology, (French) [Galois cohomology] Fifth edition. Lecture Notes in Mathematics, 5. Springer-Verlag, Berlin, 1994 [Sz] W. Szmielew, Elementary properties of Abelian groups, Fund. Math. 41 (1955), 203-271

[Wi] J. S. Wilson, Proﬁnite groups, London Mathematical Society Mono- graphs. New Series, 19. The Clarendon Press, Oxford University Press, New York, 1998

[Z] M. Ziegler, Model Theory of Modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149-213

98 Index

AC , 1 C³, 23 f , 1 ∼, 23 |A [G : H], 1 J³, 23 [x, y], 1 [M] ∨ [N], 24 K×, 2 [M] ∧ [N], 24 Ka, 2 [[x]:[y]], 24 Ks, 2 b, 25 ≤, 2 G-extension, 46 C, 2 (G, χ), 52 (G ∗ G , χ ∗ χ ), 53 Co, 2 1 2 1 2 ab X, 2 G , 62 hXi, 2 AG, 37 ≤n h|X|i, 2 AG , 38 =n [G, G], 3 AGp , 38 Gp, 3 AGp, 38 ∗ (free product), 5 AnnA∗ (B), 7 nA, 6 (AS), 39 n A[n], 6 B (G, A), 8 A∗, 6 Br(K), 10 ϕ∗, 6 Br(L/K), 10 ∗ (Pontryagin dual), 6 C, 1 AG, 7 char(K), 1 n [A], 10 C (G, A), 7 nBr(K), 10 (D, χD), 53 D, 53 Ka, 10 ∂ , 7 (a, b, ζn), 10 n (a, b), 10 Divp(A), 15 K(p), 11 E(A), 15 (L, G, f), 12 e(L/K), 19 A(α), 13 fa,b, 13 Aα, 13 f(L/K), 19 v, 14 Φ(G), 3 i A+, 17 Φ (G), 3 i (K, v), 17 F (G), 3 ϕ , 18 Kv, 18 v F , 1 KT , 19 q G, 54 KZ , 19 Gm(A, B), 28 KV , 19 (Kh, vh), 20 GK , 2 (G (p), χ ), 52 Mj, 21 K K GK (p), 11

99 (G(T, p), χ(T, p)), 56 Th(A), 1 Gal(L/K), 2 ThL(A), 1 m,s Gbd (A, B), 79 TIS, 24 m,s Gbd (AC0 , BC0 ), 79 TK, 31 m,s GIS (G, H), 28 Tor(A), 13 m,s GIS (S(G)c,S(H)c), 28 (T, p), 56 Gn, 54 up,α(A), 14 n H (G, A), 8 Up(A), 85 Homc(G, A), 8 up(G), 47 hp(G), 46 V , 19 I(χ), 65 Vp(A), 90 idA, 1 (W), 51 im(ϕ), 1 Z, 19

KFR, 45 Z, 53

K¬FR, 45 Z/n, 1 ker(ϕ), 1 Zn(G, A), 8 LC , 1 (Zp oχ G, χ), 53 LIS, 23 (Zp, χi), 53 lv, 15 ζpn , 2 Ma(T, p), 58 Zp, 1 ModP , 31 µn(K), 2 absolute pro-p Galois group, 11 µp∞ = µp∞ (C), 2 Annihilator, 7 µp∞ (K), 2 axiomatizable property of an L-structure, Mv, 18 31 N, 77 Azumaya-algebra, 9 N≥1, 52 Brauer group, 10 N≥1, 52 n(G), 63 character (of a cyclotomic pair), 52 N , 10 a character group, 17 N , 2 L/K classy game, 28 ord(g), 1 coboundary, 8 ord (g), 1 G cochain, 7 O , 17 v cocycle, 8 P, 1 cohomology group, 8 P, 54 compatible pair of maps, 8 P , 55 D complete inverse system of a group, 23 P , 54 n continuous quotient, 2 P , 55 Z converging to 1, 3 π , 24 xy crossed product, 12 q(K), 65 cup product, 9 rank(G), 3 cyclotomic res , 10 L/K p-extension, 46 S, 54 maximal, 46 S(G), 23 p-quotient, 46 S (G), 23 k maximal, 46 S , 54 n pair, 52 T , 19 attached to(T, p), 57 T , 37 AG basic, 53 Tfp(A), 15

100 product inertia free, 53 field, 19 semidirect, 53 group, 19 inertia degree, 19 DAT, 6 IS, 25 dead end, 69 of a field, 47 κ-torsion-saturated, 94 of a profinite group, 47 decomposition logical value, 15 field, 19 ÃLos’ theorem, 22 group, 19 Demushkin field, 65 macroequivalent, 57 Demushkin group, 63 macrostructure, 58 derived subgroup, 3 maximal Galois p-extension, 11 discrete group, 6 microstructure, 58 divisible, 14 N-sequence, 77 p-divisible, 14 adequate, 85 EF-invariants, 15 equivalent, 77 elementary, 15 finite, 77 Ehrenfeucht-Fraisségame, 28 infinite, 77 elementary class of L-structures, 31 norm rest algebra, 10 Elementary Type Conjecture, 52, 54 p-divisible, 14 ETC, 52, 54 p-group, 14 extension of valued fields, 18 p-pure, 14 finite exponent, 6 p-part, 14 formally real field, 45 p-adically closed field, 65 Frattini p-adic Galois group, 66 quotient, 3 p-adic Demushkin group, 67 series, 3 PAG, 6 subgroup, 3 place, 18 free presentation (of a profinite group), Poincarégroup, 74 3 Pontryagin duality, 6 free pro-p-product, 4 pro-C-group, 2 free product, 5 proabelian group, 6 PROFIN, 25 Galois pair, 52 profinite group, 2 Gauss extension, 18 profinite invariants, 90 projective profinite group, 34 height Prufer¨ group, 2 abelian p-height of G, 47 pure, 14 cyclotomic p-height of K, 47 p-pure, 14 infinite p-height, 14 infinite height, 14 ramification henselian valuation, 20 field, 19 henselization, 20 group, 19 index, 19 immediate extension, 19 ramified extension, 19 independent valuations, 18 rank of a profinite group, 3 inert extension, 19 reduced group, 14

101 relator rank, 8 residue ﬁeld, 18 restriction map between Brauer groups, 10 of cohomology groups, 9 Ryll-Nardzewski Theorem, 22 saturated structure, 22 small structure, 22 splitting ﬁeld, 10 stable, 22 κ-stable, 22 substructure of an inverse system, 25 tame extension, 19 through road, 69 torsion type, 94 tree structure, 56

Ulm invariant, 14 valuation, 17 valuation ring, 17 value group, 17 valued ﬁeld, 17 winning strategy, 28

102