DOCSLIB.ORG

1. Galois Theory 2. Kummer Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1. Galois Theory 2. Kummer Theory CLASS FIELD THEORY NOTES HAO (BILLY) LEE Abstract. These are notes I took in class, taught by Professor Frank Calegari. I claim no credit to the originality of the contents of these notes. Nor do I claim that they are without errors, nor readable. 1. Galois Theory Let L=K be a nite Galois extension with Galois group G. Then L is a K[G]-module. Problem 1.1. What is L as a K[G]-module? Claim 1.2. It's just IndhGi or just . 1 1 K [G] Proof. By primitive element theorem, L = K[x] = K(α) for some α 2 L. 2. Kummer Theory Understanding Z=pZ extensions for a prime p. Case : contains a -th root of unity . Let be a -extension with Galois group . Since ∼ , 1 K p ζp L=K Z=pZ G = hσi L = K[G] we know the map σ : L ! L has characteristic polynomial xp − 1 and has all primitive roots of unity as eigenvalues (also from σ −1 6= 0). Since ζp 2 K, we know that σ −ζp has a non-trivial kernel. That is, there exists θ 2 L such that σθ = ζpθ for some θ 6= 0. p p p p σθ = (σθ) = (ζpθ) = θ : p 1=p p Then we get θ 2 K and θ2 = K and so L = K(θ) = K γ where γ = θ 2 K. This means that every Z=pZ-extension is adjoining a p-th root. Constructive pseudo-proof: Take a random element z 2 L. If z 6= 0 and σz = ζpz then we are done. Else, let −1 −2 2 1−p p−1 y = z + ζp σz + ζp σ z + ::: + ζp σ z then σy = ζy: Just have to make sure y 6= 0. × Case 2: K general (charK 6= p). Since the Galois group of K(ζ)=K is a subgroup of (Z=pZ) which is prime to p, we see that is still a extension. Therefore, we can reduce the problem to the previous case. L(ζp)=K(ζp) Z=pZ Let be the absolute Galois group of . We want to understand . GK K hom (GK ; Z=pZ) Claim 2.1. There exists an isomorphism (assuming ζp 2 K) × ×p ∼ ∼ K =K = hom (GK ; µp) = hom (GK ; Z=pZ) : However, the last isomorphism depends on the choice of root of unity (that maps to 1). The map is σ(γ1=p) γ 7! σ 7! 2 µ : γ1=p p Proof. We have exact sequence × p × 1 ! µp ! K¯ ! K¯ ! 1: 1 CLASS FIELD THEORY NOTES 2 Take invariance under GK . This gives × p × 1 1 × K ! K ! H (GK ; µp) ! H GK ; K¯ [p] ! 0: Since 1 ¯ × by Hilbert's theorem . Therefore, we have our desired isomorphism. H GK ; K = 0 90 In general (ζp 2= K), K(ζ )× hom (G ; =p ) ,! hom G ; =p ∼ hom G ; µ ∼ p : K Z Z K(ζp) Z Z = K(ζp) p = ×p K(ζp) Which Galois extensions are possible? We know that 1 ! GK(ζp) ! GK ! G ! 1 and the action of on is just lift then conjugate. Since elements of are -invariant, everything G GK(ζp) hom (GK ; Z=pZ) G lands in the -invariant (here −1 ). The next map however is not -equivariant. We G hom GK(ζp); Z=pZ g (x) = (gxg ) G know that × and acts by a. Then the image in are just the elements where G ⊆ (Z=pZ) a 2 G [a] ζ = ζ hom GK(ζp); µp [a] c = ac. a Claim 2.2. × ×p [a]x=x . hom (GK ; Z=pZ) = (K(ζp) =K(ζp) ) Example 2.3. Let K = Q and p = 3. p p × × ×3 −1 −1 We know that G = (Z=3Z) = {±1g = hσi : Then we want x 2 Q −3 =Q −3 where σx = x (ie. σx = x modulo cubes). For example, take −1 ( is of degree , then take invariance under conjugation). Then L = Q ζ9 + ζ9 Q (ζ9) 6 1 1 −1 3 3 Q(ζ9) = Q ζ9 + ζ9 ; ζ3 = Q ζ3; γ = Q ζ3; ζ3 : Now try −1 1=3 Q ζ7 + ζ7 ; ζ3 = Q ζ3; γ : We need 1=3 −1 −1 2 2 (so the form −1 as before). γ = ζ7 + ζ7 + ζ3 ζ7 + ζ7 z + ζ σz Now take the base eld to be some Q(t). One degree 3 extension is K(t) where K=Q is degree 3 (it's usually called the isotrivial extension. The way to think about it, is that if degree 3 extensions are viewed as a family, you can have a curve that curls over 3 times, or 3 lines that look the same). p p 3 3 2.1. S3-extension of F . If n 2 F , and −3 2= F , then x − n (if n is not a cube) has splitting eld F ( n; ζ3) has Galois group S3. These all have the property that the quadratic subeld is F (ζ3). Does there exists an S3 extension of F with arbitrary quadratic extension E=F . We have extensions p 3 L L(ζ3) E(ζ3; α) Z=3Z E E(ζ3) Z=2Z F p p 3 the extension 3 is given by σp α ∼ in Galois cohomology. E(ζ3; α)=E(ζ3) σ 7! 3 2 µ3 = Z=3Z p p α p p p p p p 3 3 When does E(ζ3)( α) = E (ζ3) β ? Well, if E( α) = E( β) then σ α = − α and σ β = − β so σ acts p p p α σ 3 α 3 trivially on p so it is in the base eld. Similarly, p = ζ and σp β = ζ or ζ2. Then our condition is either β 3 α 3 β p p 3 α 3 α2 σ xes p or p : 3 β 3 β This means that α = β · cube or α = β2 · cube. CLASS FIELD THEORY NOTES 3 We know that extensions of are parametrized by × ×3. Those that come from extensions Z=3Z E(ζ3) E(ζ3) =E(ζ3) Z=3Z of are those such that Gal × acts by a. E G = (E(ζ3)=E) ,! (Z=3Z) [a] x = x We know that Gal (E(ζ3)=F ) = Gal (F (ζ3)=F ) × Gal (E=F ) when E 6= F (ζ3). × ×3 Claim 2.4. An S3-extension of F contains a quadratic eld E 6= F (ζ3) i there is a line in E(ζ3) =E(ζ3) such that [a] 2 G acts by [a] x ≡ x−1 mod cubes. p p p Alternatively, if 3 is Galois, with Galois groups . Then Gal 2. 3 3 E (ζ3; α) =F S3 × Z=2Z (E(ζ3)=F ) = (Z=2Z) α ! β 2 E(ζ3) with some β conjugate to β. Example 2.5. extensions of S3 Q p Let . Special for (and doesn't happen for . 2 acts on × ×3, and want E = Q D p = 3 p > 3) ∆ = (Z=2Z) E(ζ3) =E(ζ3) × ×3 to nd x 2 E(ζ3) =E(ζ3) such that it is an eigenvector for ∆. Write ∆ = hσ1i × hσ2i. Let x~ be any lift, then x~ σ2x~ xσ~ 1σ2x~ (1 − [σ1]) (1 − [σ2]) x = ÷ = : σ1x~ σ2σ1x~ σ1xσ~ 2x~ p Then -extensions of containing (for square) are S3 Q Q D D 6= −3 · p p p 3 L(ζ3) = Q D; −3; x p where y , is not a cube. We identify -invariant under , so x = σy y 2 Q −3D L G p r y rσy L = D; 3 + 3 : Q σy y For example, taking the splitting eld of 3 3 1 and 3 y σy . x − 3x = θ + θ3 x − 3x = σy + y p Example 2.6. Let and . Suppose . Then . Then extensions correspond to F = Fp E = Fp2 ζ3 2= F E = Fp −3 S3 G=triv [−1]x=x−1 × ×3 and extensions correspond to × ×3 By looking at the order (without descending Fp2 =Fp2 Z=6Z Fp2 =Fp2 : to eigenspace) there are possibilities, and so just unique extension. Z=3Z 1 S3 σx=x−1 σx=x−1 Now suppose . Suppose . Then extensions correspond to × ×3 ∼ × ×3 ζ3 2 F p ≡ 1 mod 3 S3 Fp2 =Fp2 = Fp2 =Fp2 = trivial. Example 2.7. extension of and (so ). S3 Qp p ≡ 1 mod 3 ζ3 2 Qp p The quadratic extensions are are either p or p . We want to pick so that Qp Qp p ; Qp ( u) Qp up E σx=x−1 E×=E×3 is non-trivial: Recall that × ∼ × Z ∼ Z E = OE × πE = (1 + πEOE) × µE × πE: Then × E 2 (1 + mE) ×3 = µE/µE × 3 × Z=3Z: E (1 + mE) Since is cyclic of order divisible by , the quotient is isomorphic to . The expression of cubes can be written as µE 3 Z=3Z a power series, which will converge on 1 + mE. That is, 1=3 X 1=3 1 + uπk = unπkn : E E n Therefore, E× =∼ ( =3 )2 : E×3 Z Z Now, we need to see how Galois acts. In the Z , the Galois will change uniformizers, but the valuation does not change. πE That is, it does nothing to the quotient Z=3Z. CLASS FIELD THEORY NOTES 4 The is either × or × . In the rst case, Galois acts trivially because there's no new roots of unity. In the section µE Fp Fp2 case, we did in the previous example and found that the Galois group acts trivially. Hence, Galois acts trivially on × ×3. Therefore, there are no extensions of . E =E S3 Qp Example 2.8.
Load more

Recommended publications
  • Open Problems: CM-Types Acting on Class Groups
    Open problems: CM-types acting on class groups Gabor Wiese∗ October 29, 2003 These are (modified) notes and slides of a talk given in the Leiden number theory seminar, whose general topic are Shimura varieties. The aim of the talk is to formulate a purely number theoretic question (of Brauer-Siegel type), whose (partial) answer might lead to progress on the so-called André-Oort conjecture on special subvarieties of Shimura varieties (thus the link to the seminar). The question ought to be considered as a special case of problem 14 in [EMO], which was proposed by Bas Edixhoven and which we recall now. Let g be a positive integer and let Ag be the moduli space of principally polarized abelian varieties of dimension g. Do there exist C; δ > 0 such that for every CM-point x of Ag (i.e. the corresponding abelian variety has CM) one has G :x C discr(R ) δ; j Q j ≥ j x j where Rx is the centre of the endomorphism ring of the abelian variety correspond- ing to x??? What we will call the Siegel question in the sequel, is obtained by restricting to simple abelian varieties. In the talk I will prove everything that seems to be known on the question up to this moment, namely the case of dimension 2. This was worked out by Bas Edixhoven in [E]. The arguments used are quite “ad hoc” and I will try to point out, where obstacles to gener- alisations seem to lie. The boxes correspond more or less to the slides I used.
    [Show full text]
  • Background on Local Fields and Kummer Theory
    MATH 776 BACKGROUND ON LOCAL FIELDS AND KUMMER THEORY ANDREW SNOWDEN Our goal at the moment is to prove the Kronecker{Weber theorem. Before getting to this, we review some of the basic theory of local fields and Kummer theory, both of which will be used constantly throughout this course. 1. Structure of local fields Let K=Qp be a finite extension. We denote the ring of integers by OK . It is a DVR. There is a unique maximal ideal m, which is principal; any generator is called a uniformizer. We often write π for a uniformizer. The quotient OK =m is a finite field, called the residue field; it is often denoted k and its cardinality is often denoted q. Fix a uniformizer π. Every non-zero element x of K can be written uniquely in the form n uπ where u is a unit of OK and n 2 Z; we call n the valuation of x, and often denote it S −n v(x). We thus have K = n≥0 π OK . This shows that K is a direct union of the fractional −n ideals π OK , each of which is a free OK -module of rank one. The additive group OK is d isomorphic to Zp, where d = [K : Qp]. n × ∼ × The decomposition x = uπ shows that K = Z × U, where U = OK is the unit group. This decomposition is non-canonical, as it depends on the choice of π. The exact sequence 0 ! U ! K× !v Z ! 0 is canonical. Choosing a uniformizer is equivalent to choosing a splitting of this exact sequence.
    [Show full text]
  • 20 the Kronecker-Weber Theorem
    18.785 Number theory I Fall 2016 Lecture #20 11/17/2016 20 The Kronecker-Weber theorem In the previous lecture we established a relationship between finite groups of Dirichlet characters and subfields of cyclotomic fields. Specifically, we showed that there is a one-to- one-correspondence between finite groups H of primitive Dirichlet characters of conductor dividing m and subfields K of Q(ζm)=Q under which H can be viewed as the character group of the finite abelian group Gal(K=Q) and the Dedekind zeta function of K factors as Y ζK (x) = L(s; χ): χ2H Now suppose we are given an arbitrary finite abelian extension K=Q. Does the character group of Gal(K=Q) correspond to a group of Dirichlet characters, and can we then factor the Dedekind zeta function ζK (S) as a product of Dirichlet L-functions? The answer is yes! This is a consequence of the Kronecker-Weber theorem, which states that every finite abelian extension of Q lies in a cyclotomic field. This theorem was first stated in 1853 by Kronecker [2] and provided a partial proof for extensions of odd degree. Weber [6] published a proof 1886 that was believed to address the remaining cases; in fact Weber's proof contains some gaps (as noted in [4]), but in any case an alternative proof was given a few years later by Hilbert [1]. The proof we present here is adapted from [5, Ch. 14] 20.1 Local and global Kronecker-Weber theorems We now state the (global) Kronecker-Weber theorem.
    [Show full text]
  • Algebraic Number Theory II
    Algebraic number theory II Uwe Jannsen Contents 1 Infinite Galois theory2 2 Projective and inductive limits9 3 Cohomology of groups and pro-finite groups 15 4 Basics about modules and homological Algebra 21 5 Applications to group cohomology 31 6 Hilbert 90 and Kummer theory 41 7 Properties of group cohomology 48 8 Tate cohomology for finite groups 53 9 Cohomology of cyclic groups 56 10 The cup product 63 11 The corestriction 70 12 Local class field theory I 75 13 Three Theorems of Tate 80 14 Abstract class field theory 83 15 Local class field theory II 91 16 Local class field theory III 94 17 Global class field theory I 97 0 18 Global class field theory II 101 19 Global class field theory III 107 20 Global class field theory IV 112 1 Infinite Galois theory An algebraic field extension L/K is called Galois, if it is normal and separable. For this, L/K does not need to have finite degree. For example, for a finite field Fp with p elements (p a prime number), the algebraic closure Fp is Galois over Fp, and has infinite degree. We define in this general situation Definition 1.1 Let L/K be a Galois extension. Then the Galois group of L over K is defined as Gal(L/K) := AutK (L) = {σ : L → L | σ field automorphisms, σ(x) = x for all x ∈ K}. But the main theorem of Galois theory (correspondence between all subgroups of Gal(L/K) and all intermediate fields of L/K) only holds for finite extensions! To obtain the correct answer, one needs a topology on Gal(L/K): Definition 1.2 Let L/K be a Galois extension.
    [Show full text]
  • Number Crunching Vs. Number Theory: Computers and FLT, from Kummer to SWAC (1850–1960), and Beyond
    Arch. Hist. Exact Sci. (2008) 62:393–455 DOI 10.1007/s00407-007-0018-2 Number crunching vs. number theory: computers and FLT, from Kummer to SWAC (1850–1960), and beyond Leo Corry Received: 24 February 2007 / Published online: 16 November 2007 © Springer-Verlag 2007 Abstract The present article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT, as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer’s contributions and its immediate influence, I present work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth- pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. I show how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large. Communicated by J.J.
    [Show full text]
  • Kummer Theory
    Lecture 12: Kummer Theory William Stein Feb 8, 2010 1 Kummer Theory of Fields Kummer theory is concerned with classifying the abelian extensions of exponent n of a field K, assuming that K contains the nth roots of unity. It's a generalization of the correspondence between quadratic extensions of Q and non-square squarefree integers. Let n be a positive integer, and let K be a field of characteristic prime to n. Let L be a separable closure of K. Let µn(L) denote the set of elements of order dividing n in L. Lemma 1.1. µn(L) is a cyclic group of order n. n Proof. The elements of µn(L) are exactly the roots in L of the polynomial x −1. Since n n is coprime to the characteristic, all roots of x − 1 are in L, so µn(L) has order at n least n. But K is a field, so x − 1 can have at most n roots, so µn(L) has order n. Any finite subgroup of the multiplicative group of a field is cyclic, so µn(L) is cyclic. Consider the exact sequence n ∗ x 7! x ∗ 1 ! µn(L) ! L −−−−−! L ! 1 of GK = Gal(L=K)-modules. The associated long exact sequence of Galois cohomology yields ∗ ∗ n 1 1 ∗ 1 ! K =(K ) ! H (K; µn(L)) ! H (K; L ) !··· : We proved that H1(K; L∗) = 0, so we conclude that ∗ ∗ n ∼ 1 K =(K ) = H (K; µn(L)); where the isomorphism is via the δ connecting homomorphism. If α 2 L∗, we obtain the 1 ∗ n corresponding element δ(α) 2 H (K; µn(L)) by finding some β 2 L such that β = α; then the corresponding cocycle is σ 7! σ(β)/β 2 µn(L).
    [Show full text]
  • Cyclic Galois Extensions of Fixed Degree P" in Detail
    TRANSACTIONSOF THE AMERICAN MATHEMATICALSOCIETY Volume 326, Number 1, July 1991 CYCLICGALOIS EXTENSIONS AND NORMAL BASES C. GREITHER Abstract. A Kummer theory is presented which does not need roots of unity in the ground ring. For R commutative with p~ e R we study the group of cyclic Galois extensions of fixed degree p" in detail. Our theory is well suited for dealing with cyclic p"-extensions of a number field K which are unramified outside p. We then consider the group Gal(tfK[p~ ], C' n) of all such extensions, and its subgroup NB^tp- ], C«) of extensions with integral normal basis outside p . For the size of the latter we get a simple asymptotic formula (n —►oo), and the discrepancy between the two groups is in some way measured by the defect ô in Leopoldt's conjecture. Introduction This work begins with a general study of Galois extensions of a commutative ring R with finite abelian Galois group G (Part I). The results are applicable in a number-theoretical setting (Part II) and give theorems concerning the existence of //-integral normal bases in C »-extensions and Z -extensions of an arbitrary number field K . (Here p is an odd prime , C » is the cyclic group of order p" , and "//-integral" means "integral outside places over p ".) Let us start by describing the general situation. Let R be a commutative ring with 1 and G a finite abelian group. We study the finite abelian group Gal(i?, G) which consists of the (isomorphism classes of) G-Galois ring exten- sions of R.
    [Show full text]
  • Kummer, Regular Primes, and Fermat's Last Theorem 1
    KUMMER, REGULAR PRIMES, AND FERMAT'S LAST THEOREM ILA VARMA CALIFORNIA INSTITUTE OF TECHNOLOGY Abstract. This paper rephrases Kummer's proof of many cases of Fermat's last theorem in contemporary notation that was in fact derived from his work. Additionally, it develops a reformulation of the proof using class field theory from a modern perspective in a manner similar to the tactics used for the complete proof, and describes how Kummer's proof strategy can generalize to solve the theorem for a broader set of primes. 1. Introduction Ernst Kummer was a 19th century mathematician who came across Fermat's last theorem in attempts to generalize the law of quadratic reciprocity and study higher reciprocity laws. While he described those as \the principal subject and the pinnacle of contemporary number theory," he considered Fermat's last theorem a \curiosity of number theory rather than a major item" [1]. A priori, this was not an unreasonable opinion of a problem that could be understood by a 12-year-old. We state this mere curiosity below. Theorem. For any integer n > 2, the equation xn + yn = zn has no non-trivial solutions in the integers, i.e. if x; y; z 2 Z satisfy this equation, then xyz = 0. Despite his disinterest, Kummer made the first substantial step in proving a part of Fermat's last theorem for many cases. This came only a few weeks after Gabriel Lam´eincorrectly announced that he had found a complete proof [1]. Lam´edid make the breakthrough in attempting to decompose xn + yn into linear factors by introducing the complex numbers satisfying ζn = 1, known today as roots of unity.
    [Show full text]
  • Galois Cohomology Summary
    Galois Cohomology Summary Thomas Browning, Thomas Carr April 2019 1 Kummer Theory An abelian extension L=K with G = Gal(L=K) is said to be of exponent n if σn = 1 for all σ 2 G.A Kummer extension is an abelian extension of exponent n where K contains the nth roots of unity. 3 Example: Let L be the splitting field of x − a over Q where a is not a cube, and let K = Q(ζ3). Then L=K is a Kummer extension of exponent 3. Let L=K be a Kummer extension of exponent n, and consider the exact sequence × ·n × n 1 µn L (L ) 1 Taking Galois cohomology we have the long exact sequence × ·n × × n 1 1 × 1 µn K K \ (L ) H (G; µn) H (G; K ) ··· 1 × We have H (G; K ) = 1 by Hilbert's Theorem 90. Since G acts trivially on µn, we also have that 1 × × × n × n H (G; µn) = HomZ(G; µn). Because im(K −! K \ (L ) ) = (K ) , we have an isomorphism × × n × n (K \ (L ) )=(K ) ' HomZ(G; µn): n Since σ = 1 for all σ 2 G, the abelian group HomZ(G; µn) is just the Pontryagin dual χ(G) of G. Pontryagin duality gives us an inclusion reversing isomorphism of subgroup lattices of χ(G) and G. By Galois Theory, subgroups of G correspond to finite Kummer extensions of K. Thus we have a correspondence between subgroups of K× \ (L×)n containing (K×)n and Kummer extensions of K contained in L. It can be shown that there exists a maximal Kummer extension N containing all other Kummer extensions of exponent n.
    [Show full text]
  • Kummer Type Extensions in Function Fields
    International Journal of Algebra, Vol. 7, 2013, no. 4, 157 - 166 HIKARI Ltd, www.m-hikari.com Kummer Type Extensions in Function Fields Marco S´anchez-Mirafuentes Universidad Aut´onoma Metropolitana Unidad Iztapalapa Divisi´on de Ciencias B´asicas e Ingenier´ia Departamento de Matem´aticas Av. San Rafael Atlixco 186, Col Vicentina Del. Iztapalapa, C.P. 09240 M´exico, D.F. Gabriel Villa Salvador CINVESTAV IPN Departamento de Control Autom´atico Apartado Postal 14-740 M´exico, D.F. C.P 07000 Copyright c 2013 Marco S´anchez-Mirafuentes and Gabriel Villa Salvador. This is an open access article distributed under the Creative Commons Attribution License, which per- mits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We present a generalization of Kummer extensions in algebraic func- tion fields with finite field of constants Fq, using the action of Carlitz- Hayes. This generalization of Kummer type extensions is due to Wen- Chen Chi and Anly Li and due also to Fred Schultheis. The main results of this article are Proposition 3.2 and Theorem 3.4. They provide a partial analogue of a theorem of Kummer, which establishes a bijection between Kummer extensions L/K of exponent n and subgroups of K∗ containing (K∗)n. Mathematics Subject Classification: 11R60, 11R99, 12F05, 14H05 Keywords: Carlitz Module, Kummer extensions, Dual module, Bilinear map 158 M. S´anchez-Mirafuentes and G. Villa-Salvador 1 Introduction An n-Kummer extension, is a field extension L/K such that n is relatively prime to char(K), the characteristic of K, μn ⊆ K, where μn is the group of n-th roots of unity, and √ L = K( n Δ), ∗ ∗n where√ Δ is a subgroup of K containing the group√ K of n-th powers, and K( n Δ) is the field generated by all the roots n a, with a ∈ Δ.
    [Show full text]
  • Cohomology of Number Fields, by Jürgen Neukirch, Alexander
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 1, Pages 101{107 S 0273-0979(01)00924-7 Article electronically published on October 10, 2001 Cohomology of number fields,byJ¨urgen Neukirch, Alexander Schmidt, and Kay Wingberg, Grundlehren der mathematischen Wissenschaften, vol. 323, Springer- Verlag, 2000, 720 pp., $109.00, ISBN 3-540-66671-0 The story begins in the mid-1930s, with the work of Witold Hurewicz on alge- braic topology. More specifically, the question was how the fundamental group of a complex was related to its homology and cohomology groups. After defining the higher homotopy groups in 1935, Hurewicz considered aspherical complexes, that is, complexes where all higher homotopy groups are zero. For such complexes, he proved in [10] that all the homology and cohomology groups were entirely deter- mined by the fundamental group. At that point, it was natural to ask whether one could find an algebraic descrip- tion of the connection between the fundamental group and the homology groups, one that did not depend on considering the topological space at all. A first step in this direction came from a theorem of Heinz Hopf in [9] that gave an explicit way to construct the second homology group of an aspherical complex in terms of its fundamental group. The construction is non-obvious, and at first glance appears to be non-canonical, since it is described in terms of a presentation of the funda- mental group by generators and relations. Hopf proved that in fact the group he constructed did not depend on the choice of presentation, suggesting that there must be a more \invariant" way of describing it.
    [Show full text]
  • Class Field Towers
    Class Field Towers Franz Lemmermeyer September 7, 2010 Contents 1 Some History 1 1.1 The Prehistory . 1 1.1.1 Fermat, Euler, and Gauß: Binary Quadratic Forms . 1 1.1.2 Kummer and Dedekind: Ideals . 2 1.1.3 Kummer and Weber: Kummer Theory . 3 1.2 The Genesis of the Hilbert Class Field . 5 1.2.1 Kronecker and Weber: Complex Multiplication . 5 1.2.2 Hilbert: Hilbert Class Fields . 6 1.2.3 Furtw¨angler: Singular Numbers . 6 1.2.4 Takagi: Class Field Theory . 7 1.2.5 From Hilbert to Hasse . 8 1.3 Genus Class Fields . 9 1.3.1 Quadratic Number Fields . 9 1.3.2 Abelian Number Fields . 10 1.3.3 General Number Fields . 10 1.3.4 Central Extensions . 11 1.4 2-Class fields . 15 1.4.1 Quadratic Number Fields . 15 1.4.2 Cubic Number Fields . 18 1.4.3 General Number Fields . 19 1.5 3-Class Fields . 19 1.5.1 Quadratic Number Fields . 19 1.5.2 Cubic Fields . 21 1.6 `-Class Fields . 22 1.6.1 Quadratic Number Fields . 22 1.6.2 Cyclic Number Fields . 22 1.7 Separants . 22 1.8 Capitulation of Ideal Classes . 23 1.8.1 Hilbert’s Theorem 94 . 23 1.8.2 Artin’s Reduction . 24 1.8.3 Scholz and Taussky . 25 1.8.4 Principal Ideal Theorems . 30 1.9 Class Field Towers . 31 1.9.1 Terminating Class Field Towers . 31 1.9.2 Golod-Shafarevic: Infinite Class Field Towers . 35 1.9.3 Odlyzko Bounds .
    [Show full text]