DOCSLIB.ORG

Recent Progress in Determining P-Class Field Towers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Recent Progress in Determining P-Class Field Towers Recent Progress in Determining p-Class Field Towers Conference: 1st International Colloquium ANCI 2016 Place: Facult´ePolydisciplinaire Taza, USMBA Venue: Taza, Morocco Date: November 11 { 12, 2016 Author: Daniel C. Mayer (Graz, Austria) Affiliation: Austrian Science Fund (FWF) Keynote within the frame of the international scientific research project Towers of p-Class Fields over Algebraic Number Fields supported by the Austrian Science Fund (FWF): P 26008-N25 1 2 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 INTRODUCTION. • The key for determining the Galois group (1) (1) G := Gp F = Gal(Fp =F ) (1) of the unramified Hilbert p-class field tower Fp , i.e. the maximal unramified pro-p extension, of an algebraic number field F is the Artin pattern AP(G) combined with bounds for the relation rank d2G. G is briefly called the p-tower group of F . • My principal goal is to draw the attention of the audience to the striking novelty of three-stage towers of p-class fields over quadratic, cubic and quartic number fields F , discovered by myself in the past four years, partially in cooperation with M. R. Bush (WLU, VA) and M. F. Newman (ANU, ACT). • This lecture can be downloaded from http://www.algebra.at/ANCI2016DCM.pdf • It is an updated and compact version of my article [9] D. C. Mayer, Recent progress in determining p-class field towers, Gulf J. Math. (Dubai, UAE), arXiv: 1605.09617v1 [math.NT] 31 May 2016. 3 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 1. The Hilbert p-class field tower Assumptions: p : : : a prime number, F::: an algebraic number field, ClpF := SylpClF ::: the p-class group of F . (n) Definition. For n ≥ 0, the nth Hilbert p-class field Fp is the maximal unramified Galois extension of F with group n (n) Gp F := Gal Fp =F n having derived length dl(Gp F ) ≤ n and order a power of p. (1) The Hilbert p-class field tower Fp over F is the maximal unramified pro-p extension of F and has the Galois group 1 (1) n Gp F := Gal Fp =F ' lim− Gp F: n≥0 (1) (2) (n) (1) F ≤ Fp ≤ Fp ≤ ··· ≤ Fp ≤ ··· ≤ Fp 1 GpF 2 GpF 2nd p-class group n Gp F nth p-class group 1 G := Gp F p-tower group 4 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 n The nth stage Gp F of the p-class tower arises as the nth derived quotient G=G(n) of the p-tower group G, for n ≥ 1. In particular, for n = 2, the second p-class group 2 (2) M := GpF ' G=G can be viewed as a two-stage approximation of G. (1) (2) (n) (1) F ≤ Fp ≤ Fp ≤ ··· ≤ Fp ≤ ··· ≤ Fp G=G0 2 00 GpF ' G=G 2nd p-class group n (n) Gp F ' G=G nth p-class group 1 G = Gp F p-tower group Definition. The derived length dl(G) of the p-tower group 1 G = Gp F is called the length `pF of the p-class tower of F . (1) When do we consider the p-class tower Fp as \known"? Is it sufficient to determine its length `pF ? No, not at all! The length alone is a poor amount of informa- tion. According to my conviction, we are not done before we can give an explicit pro-p presentation of the p-tower group 1 (1) G = Gp F = Gal Fp =F : 5 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 An equivalent inductive definition of the p-class tower: (1) S (i) The tower Fp := i≥0 Fp arises recursively from the base (0) field Fp := F by the successive construction of maximal (1) (n) (n−1) abelian unramified p-extensions Fp := Fp , n ≥ 1. p By the Artin reciprocity law of class field theory [1], the (n) (n−1) relative Galois group Gal Fp =Fp is isomorphic to the (n−1) (abelian!) p-class group ClpFp , for each n ≥ 1. In particular, for n = 1, we have the well-known fact that the first p-class group is isomorphic to the ordinary p-class group 1 (1) GpF = Gal Fp =F ' ClpF: In particular, for n = 2, the second p-class group (2) M = Gal Fp =F has an abelian commutator subgroup 0 (2) (1) (1) M = Gal Fp =Fp ' ClpFp and is therefore metabelian. The abelianization is 0 (1) M=M ' Gal Fp =F ' ClpF: Similarly, the abelianization of the p-tower group G is 0 (1) (1) (1) (1) G=G = Gal Fp =F =Gal Fp =Fp ' Gal Fp =F ' ClpF and consequently G is a finitely generated pro-p group. 6 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 2. Infinite p-class towers Let G be a finitely generated pro-p group with generator rank 1 2 d1 := dimFp H (G; Fp) and relation rank d2 := dimFp H (G; Fp). Theorem (Golod, Shafarevich, 1964 [2]). 2 d1−1 If G is finite, then d2 > 2 . Theorem (Refinement by Vinberg, Gasch¨utz,1965 [3]). 2 2 (d1) (d1) #G < 1 ) d2 > 4 , resp. d2 ≤ 4 ) #G = 1. Let 1 ! R ! F ! G ! 1 be a presentation of G with a free pro-p group F such that d1F = d1G, and suppose that R ⊂ Fn for a term of the Zassenhaus filtration (Fn)n≥1 of F . Theorem (Refinement by Koch, Venkov, 1975 [4]). n (d1) n−1 If G is finite, then d2 > nn · (n − 1) . Corollary (Koch, Venkov, 1975 [4]). Let p ≥ 3 be odd, then p a complex quadratic field F = Q( d) with p-class rank %p ≥ 3 has an infinite p-class tower with `pF = 1. Proof. Since F has Dirichlet unit rank r = 0, the p-tower 1 group G = Gp F has a balanced presentation with d2 = d1 and R ⊂ F3. Thus, a sufficient condition for `pF = 1 is 3 q p (d1) 2 27 3 d1 ≤ 33 · 2 , i.e. %p = d1 ≥ 4 = 2 3 ≈ 2:598. Example. d = −4 447 704 ) Cl3F ' (3; 3; 3) ) `3F = 1. However, 1 we are far from knowing a pro-3 presentation of G = G3 F . We only have the lower bound #M ≥ 317 for M = G=G00. 7 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 3. The relation rank of the p-tower group Theorem (I. R. Shafarevich, 1964 [5]). Let p ≥ 2 be a prime number, and K be an algebraic number field with signature (r1; r2) and torsion free Dirichlet unit rank r = r1 + r2 − 1, ζ a primitive pth root of unity, 1 1 G = Gp (K) = Gal(Fp (K)jK) the Galois group of the 1 maximal unramified pro-p extension Fp (K) of K, 1 d1 = dimFp H (G; Fp) the generator rank of G, 2 d2 = dimFp H (G; Fp) the relation rank of G. Then ( 0 if ζ 62 K; d ≤ d ≤ d + r + θ; where θ = 1 2 1 1 if ζ 2 K: p Corollary. K = Q( d) quadratic field with discriminant d, 1 1 G = Gp (K) = Gal(Fp (K)jK) the p-class tower group of K. Then 8 d = d if (d < 0 and p ≥ 3); <> 2 1 d1 ≤ d2 ≤ d1 + 1 if either (d < 0 and p = 2) or (d > 0 and p ≥ 3); > :d1 ≤ d2 ≤ d1 + 2 if (d > 0 and p = 2): [5] I. R. Shafarevich, Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Etudes´ Sci. 18 (1964), 71{95 (Russian). English transl. by J. W. S. Cassels: Am. Math. Soc. Transl., II. Ser., 59 (1966), 128{149. 8 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 4. Leaving the realm of class field theory p : : : a prime number, F::: an algebraic number field, E=F : : : an unramified abelian p-extension, ClpE::: the p-class group of E, TF;E : ClpF ! ClpE, a ·PF 7! (aOE) ·PE ::: the transfer of p-classes from F to E. We translate TF;E from number theory to group theory by entering the system of metabelian unramified extensions [6]: System of abelian System of metabelian unramified extensions unramified extensions (1) (1) (2) F ≤ E ≤ Fp ≤ Ep ≤ Fp Galois correspondence l l l l l M D H D M0 D H0 D 1 k k k k (2) (2) (2) (1) (2) (1) G Fp =F G Fp =E G Fp =Fp G Fp =Ep abelian quotient (1) G Ep =E 0 ' ClpE ' H=H 9 Daniel C. Mayer (Austrian Science Fund), Determining p-class towers, ANCI Taza 2016 (2) 0 (2) (1) Let H := Gal(Fp =E), then H := Gal(Fp =Ep ), and the p-class transfer TF;E is connected with the Artin transfer TM;H by two applications of the Artin reciprocity map: TF;E ClpF −! ClpE Artin isomorphism l === l Artin isomorphism M=M0 −! H=H0 TM;H In particular, we have 0 isomorphic domains, ClpF ' M=M , isomorphic kernels, ker TF;E ' ker TM;H, 0 isomorphic targets, ClpE ' H=H . Definition. Recall that the Artin transfer [6] from a group G to a subgroup H ≤ G with finite index n := (G : H) is defined with the aid of the permutation π 2 Sn of a transversal n S_ G = i=1 `i · H induced by the action of an element x 2 G: n 0 0 0 Y −1 0 TG;H : G=G ! H=H ; x · G 7! `π(i)x`i · H ; i=1 where x`i · H = `π(i) · H, for each 1 ≤ i ≤ n.
Load more

Recommended publications
  • Dirichlet Series Associated to Cubic Fields with Given Quadratic Resolvent 3
    DIRICHLET SERIES ASSOCIATED TO CUBIC FIELDS WITH GIVEN QUADRATIC RESOLVENT HENRI COHEN AND FRANK THORNE s Abstract. Let k be a quadratic field. We give an explicit formula for the Dirichlet series P Disc(K) − , K | | where the sum is over isomorphism classes of all cubic fields whose quadratic resolvent field is iso- morphic to k. Our work is a sequel to [11] (see also [15]), where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q, and in a companion paper we do the same for quartic fields having a given cubic resolvent. As an application, we compute tables of the number of S3-sextic fields E with Disc(E) < X, | | for X ranging up to 1023. An accompanying PARI/GP implementation is available from the second author’s website. 1. Introduction A classical problem in algebraic number theory is that of enumerating number fields by discrim- inant. Let Nd±(X) denote the number of isomorphism classes of number fields K with deg(K)= d and 0 < Disc(K) < X. The quantity Nd±(X) has seen a great deal of study; see (for example) [10, 7, 23]± for surveys of classical and more recent work. It is widely believed that Nd±(X)= Cd±X +o(X) for all d 2. For d = 2 this is classical, and the case d = 3 was proved in 1971 work of Davenport and Heilbronn≥ [13].
    [Show full text]
  • The Fundamental System of Units for Cubic Number Fields
    University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 2020 The Fundamental System of Units for Cubic Number Fields Janik Huth University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Other Mathematics Commons Recommended Citation Huth, Janik, "The Fundamental System of Units for Cubic Number Fields" (2020). Theses and Dissertations. 2385. https://dc.uwm.edu/etd/2385 This Thesis is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. THE FUNDAMENTAL SYSTEM OF UNITS FOR CUBIC NUMBER FIELDS by Janik Huth A Thesis Submitted in Partial Fulllment of the Requirements for the Degree of Master of Science in Mathematics at The University of Wisconsin-Milwaukee May 2020 ABSTRACT THE FUNDAMENTAL SYSTEM OF UNITS FOR CUBIC NUMBER FIELDS by Janik Huth The University of Wisconsin-Milwaukee, 2020 Under the Supervision of Professor Allen D. Bell Let K be a number eld of degree n. An element α 2 K is called integral, if the minimal polynomial of α has integer coecients. The set of all integral elements of K is denoted by OK . We will prove several properties of this set, e.g. that OK is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group × , dened as the set of all invertible elements OK of OK .
    [Show full text]
  • Number Theory
    Number Theory Alexander Paulin October 25, 2010 Lecture 1 What is Number Theory Number Theory is one of the oldest and deepest Mathematical disciplines. In the broadest possible sense Number Theory is the study of the arithmetic properties of Z, the integers. Z is the canonical ring. It structure as a group under addition is very simple: it is the infinite cyclic group. The mystery of Z is its structure as a monoid under multiplication and the way these two structure coalesce. As a monoid we can reduce the study of Z to that of understanding prime numbers via the following 2000 year old theorem. Theorem. Every positive integer can be written as a product of prime numbers. Moreover this product is unique up to ordering. This is 2000 year old theorem is the Fundamental Theorem of Arithmetic. In modern language this is the statement that Z is a unique factorization domain (UFD). Another deep fact, due to Euclid, is that there are infinitely many primes. As a monoid therefore Z is fairly easy to understand - the free commutative monoid with countably infinitely many generators cross the cyclic group of order 2. The point is that in isolation addition and multiplication are easy, but together when have vast hidden depth. At this point we are faced with two potential avenues of study: analytic versus algebraic. By analytic I questions like trying to understand the distribution of the primes throughout Z. By algebraic I mean understanding the structure of Z as a monoid and as an abelian group and how they interact.
    [Show full text]
  • P-INTEGRAL BASES of a CUBIC FIELD 1. Introduction Let K = Q(Θ
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 1949{1953 S 0002-9939(98)04422-0 p-INTEGRAL BASES OF A CUBIC FIELD S¸ABAN ALACA (Communicated by William W. Adams) Abstract. A p-integral basis of a cubic field K is determined for each rational prime p, and then an integral basis of K and its discriminant d(K)areobtained from its p-integral bases. 1. Introduction Let K = Q(θ) be an algebraic number field of degree n, and let OK denote the ring of integral elements of K.IfOK=α1Z+α2Z+ +αnZ,then α1,α2,...,αn is said to be an integral basis of K. For each prime··· ideal P and each{ nonzero ideal} A of K, νP (A) denotes the exponent of P in the prime ideal decomposition of A. Let P be a prime ideal of K,letpbe a rational prime, and let α K.If ∈ νP(α) 0, then α is called a P -integral element of K.Ifαis P -integral for each prime≥ ideal P of K such that P pO ,thenαis called a p-integral element | K of K.Let!1;!2;:::;!n be a basis of K over Q,whereeach!i (1 i n)isap-integral{ element} of K.Ifeveryp-integral element α of K is given≤ as≤ α = a ! + a ! + +a ! ,wherethea are p-integral elements of Q,then 1 1 2 2 ··· n n i !1;!2;:::;!n is called a p-integral basis of K. { In Theorem} 2.1 a p-integral basis of a cubic field K is determined for every rational prime p, and in Theorem 2.2 an integral basis of K is obtained from its p-integral bases.
    [Show full text]
  • 22 Ring Class Fields and the CM Method
    18.783 Elliptic Curves Spring 2015 Lecture #22 04/30/2015 22 Ring class fields and the CM method p Let O be an imaginary quadratic order of discriminant D, let K = Q( D), and let L be the splitting field of the Hilbert class polynomial HD(X) over K. In the previous lecture we showed that there is an injective group homomorphism Ψ: Gal(L=K) ,! cl(O) that commutes with the group actions of Gal(L=K) and cl(O) on the set EllO(C) = EllO(L) of roots of HD(X) (the j-invariants of elliptic curves with CM by O). To complete the proof of the the First Main Theorem of Complex Multiplication, which asserts that Ψ is an isomorphism, we need to show that Ψ is surjective; this is equivalent to showing the HD(X) is irreducible over K. At the end of the last lecture we introduced the Artin map p 7! σp, which sends each unramified prime p of K to the unique automorphism σp 2 Gal(L=K) for which Np σp(x) ≡ x mod q; (1) for all x 2 OL and primes q of L dividing pOL (recall that σp is independent of q because Gal(L=K) ,! cl(O) is abelian). Equivalently, σp is the unique element of Gal(L=K) that Np fixes q and induces the Frobenius automorphism x 7! x of Fq := OL=q, which is a generator for Gal(Fq=Fp), where Fp := OK =p. Note that if E=C has CM by O then j(E) 2 L, and this implies that E can be defined 2 3 by a Weierstrass equation y = x + Ax + B with A; B 2 OL.
    [Show full text]
  • 22 Ring Class Fields and the CM Method
    18.783 Elliptic Curves Spring 2017 Lecture #22 05/03/2017 22 Ring class fields and the CM method Let O be an imaginary quadratic order of discriminant D, and let EllO(C) := fj(E) 2 C : End(E) = Cg. In the previous lecture we proved that the Hilbert class polynomial Y HD(X) := HO(X) := X − j(E) j(E)2EllO(C) has integerp coefficients. We then defined L to be the splitting field of HD(X) over the field K = Q( D), and showed that there is an injective group homomorphism Ψ: Gal(L=K) ,! cl(O) that commutes with the group actions of Gal(L=K) and cl(O) on the set EllO(C) = EllO(L) of roots of HD(X). To complete the proof of the the First Main Theorem of Complex Multiplication, which asserts that Ψ is an isomorphism, we need to show that Ψ is surjective, equivalently, that HD(X) is irreducible over K. At the end of the last lecture we introduced the Artin map p 7! σp, which sends each unramified prime p of K (prime ideal of OK ) to the corresponding Frobenius element σp, which is the unique element of Gal(L=K) for which Np σp(x) ≡ x mod q; (1) for all x 2 OL and primes qjp (prime ideals of OL that divide the ideal pOL); the existence of a single σp 2 Gal(L=K) satisfying (1) for all qjp follows from the fact that Gal(L=K) ,! cl(O) is abelian. The Frobenius element σp can also be characterized as follows: for each prime qjp the finite field Fq := OL=q is an extension of the finite field Fp := OK =p and the automorphism σ¯p 2 Gal(Fq=Fp) defined by σ¯p(¯x) = σ(x) (where x 7! x¯ is the reduction Np map OL !OL=q), is the Frobenius automorphism x 7! x generating Gal(Fq=Fp).
    [Show full text]
  • The Twelfth Problem of Hilbert Reminds Us, Although the Reminder Should
    Some Contemporary Problems with Origins in the Jugendtraum Robert P. Langlands The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate devel• opments. The first of these, the theory of class fields or of abelian extensions of number fields, attained what was pretty much its final form early in this century. The second, the algebraic theory of elliptic curves and, more generally, of abelian varieties, has been for fifty years a topic of research whose vigor and quality shows as yet no sign of abatement. The third, the theory of automorphic functions, has been slower to mature and is still inextricably entangled with the study of abelian varieties, especially of their moduli. Of course at the time of Hilbert these subjects had only begun to set themselves off from the general mathematical landscape as separate theories and at the time of Kronecker existed only as part of the theories of elliptic modular functions and of cyclotomicfields. It is in a letter from Kronecker to Dedekind of 1880,1 in which he explains his work on the relation between abelian extensions of imaginary quadratic fields and elliptic curves with complex multiplication, that the word Jugendtraum appears. Because these subjects were so interwoven it seems to have been impossible to disentangle the different kinds of mathematics which were involved in the Jugendtraum, especially to separate the algebraic aspects from the analytic or number theoretic. Hilbert in particular may have been led to mistake an accident, or perhaps necessity, of historical development for an “innigste gegenseitige Ber¨uhrung.” We may be able to judge this better if we attempt to view the mathematical content of the Jugendtraum with the eyes of a sophisticated contemporary mathematician.
    [Show full text]
  • L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
    L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field.
    [Show full text]
  • Arizona Winter School 2014 Course Outline and Project Description: Asymptotics for Number Fields and Class Groups
    ARIZONA WINTER SCHOOL 2014 COURSE OUTLINE AND PROJECT DESCRIPTION: ASYMPTOTICS FOR NUMBER FIELDS AND CLASS GROUPS MELANIE MATCHETT WOOD ASSISTED BY: ROB HARRON 1. Course Outline This course will cover the questions of counting number fields and determining the distri- bution of class groups of number fields. The starting point is the very basic question: how many number fields are there? This is commonly interpreted as determining the aymptotics in X for #fK ⊂ Q¯ j jDisc(K)j ≤ Xg: A typical refinement is to determine the asymptotics for number fields with a fixed Galois group G, i.e. for #fK ⊂ Q¯ j Gal(K=Q) ' G; jDisc(K)j ≤ Xg: For class groups, what do random class groups look like? For instance, if A is a fixed odd finite abelian group and Cl(K)odd denotes the odd part of the class group of K, what proportion of quadratic fields have Cl(K)odd ' A, i.e. what is #fK ⊂ ¯ j [K : ] = 2; jDisc(K)j ≤ X; Cl(K) ' Ag lim Q Q odd ? X!1 #fK ⊂ Q¯ j [K : Q] = 2; jDisc(K)j ≤ Xg We will discuss the various heuristics and conjectures that address such questions and their generalizations, as well as the relationship between the distribution of number fields and the distribution of class groups. For two specific questions, we will go into detail to show what can actually be proven and how. We will also provide an overview of what the most recent results are in the area. Here is an outline of the course: • Introduction { State the questions of counting number fields and class groups asymptotically { Connection between counting number fields and counting class groups
    [Show full text]
  • Class Field Theory and the Local-Global Property
    Class Field Theory and the Local-Global Property Noah Taylor Discussed with Professor Frank Calegari March 18, 2017 1 Classical Class Field Theory 1.1 Lubin Tate Theory In this section we derive a concrete description of a relationship between the Galois extensions of a local field K with abelian Galois groups and the open subgroups of K×. We assume that K is a finite extension of Qp, OK has maximal ideal m, and we suppose the residue field k of K has size q, a power of p. Lemma 1.1 (Hensel). If f(x) is a polynomial in O [x] and a 2 O with f(a) < 1 then there is K K f 0(a)2 some a with f(a ) = 0 and ja − a j ≤ f(a) . 1 1 1 f 0(a) f(a) Proof. As in Newton's method, we continually replace a with a− f 0(a) . Careful consideration of the valuation of f(a) shows that it decreases while f 0(a) does not, so that f(a) tends to 0. Because K is complete, the sequence of a's we get has a limit a1, and we find that f(a1) = 0. The statement about the distance between a and a1 follows immediately. We can apply this first to the polynomial f(x) = xq − x and a being any representative of any element in k to find that K contains the q − 1'st roots of unity. It also follows that if L=K is an unramified extension of degree n, then L = K(ζ) for a qn − 1'st root of unity ζ.
    [Show full text]
  • Numerical Verification of the Cohen-Lenstra-Martinet Heuristics and of Greenberg’S P-Rationality Conjecture Razvan Barbulescu, Jishnu Ray
    Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg’s p-rationality conjecture Razvan Barbulescu, Jishnu Ray To cite this version: Razvan Barbulescu, Jishnu Ray. Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg’s p-rationality conjecture. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, In press, 32 (1), pp.159-177. hal-01534050v3 HAL Id: hal-01534050 https://hal.archives-ouvertes.fr/hal-01534050v3 Submitted on 18 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg’s p-rationality conjecture par Razvan Barbulescu et Jishnu Ray Résumé. Dans cet article nous apportons des éléments en faveur de la conjecture de Greenberg d’existence de corps p-rationnels à groupe de Galois connu. Nous intruisons une famille de corps biquadratiques p-rationnels et nous donnons des nouveaux exemples numériques de corps p-rationnels multiquadratiques de grand degré. Dans le cas des corps multiquadratiques et multicubiques on prouve que la conjec- ture est une conséquence de la conjonction de l’heuristique de Cohen- Lenstra-Martinet et d’une conjecture de Hofmann et Zhang portant sur le régulateur p-adique; nous apportons des nouveauz résultats numériques en faveur de ces conjectures.
    [Show full text]
  • Course Notes, Global Class Field Theory Caltech, Spring 2015/16
    COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 M. FLACH 1. Global class field theory in classical terms Recall the isomorphism Z Z × −!∼ Q Q 7! 7! a ( =m ) Gal( (ζm)= ); a (ζm ζm) where ζm is a primitive m-th root of unity. This is the reciprocity isomorphism of global class field theory for the number field Q. What could a generalization of this to any number field K be? Of course one has a natural map (going the other way) ∼ × Gal(K(ζm)=K) ! Aut(µm) = (Z=mZ) ; σ 7! (ζ 7! σ(ζ)) called the cyclotomic character. It is even an isomorphism if K is linearly disjoint from Q(ζm). But that's not class field theory, or rather it's only a tiny part of class field theory for the field K. The proper, and much more stunning generalisation is a reciprocity isomorphism ∼ Hm −! Gal(K(m)=K) between so called ray class groups Hm of modulus m and Galois groups of corre- sponding finite abelian extensions K(m)=K. In a first approximation one can think of a modulus m as an integral ideal in OK but it is slightly more general. The ray m × class group H isn't just (OK =m) although the two groups are somewhat related. In any case Hm is explicit, or one should say easily computable as soon as one can O× O compute the unit group K and the class group Cl( K ) of the field K. Note that both are trivial for K = Q, so that ray class groups are particularly easy for K = Q.
    [Show full text]