On a certainl-adic representation

Inventiones Mathematicae, 1978

Journal of Algebra, 1990

2016

Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain some non-trivial information about the structure of the profinite group A, for every number field, by application of results published in our book on class field theory. Résumé.-A partir d'un article de Athanasios Angelakis et Peter Stevenhagen sur la détermination de corps quadratiques imaginaires ayant le même groupe de Galois Abélien absolu A, nous étudions cette propriété pour les corps de nombres quelconques. Nous montrons qu'une telle propriété n'est probablement pas facilement généralisable, en dehors des corps quadratiques imaginaires, en raison d'obstructions p-adiques provenant des unités globales. En se restreignant aux p-sous-groupes de Sylow de A, nous montrons que l'étude correspondante est liée à une généralisation de la notion classique de corps p-rationnels. Cependant, nous obtenons des informations non triviales sur la structure du groupe profini A, pour tout corps de nombres, par application de résultats publiés dans notre livre sur la théorie du corps de classes.

2001

1992

Let K/k be a finite Galois field extension, and assume k is not an algebraic extension of a finite field. Let K* be the multiplicative group of K , and let &{K/k) be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group T = K*/Q(K/k) be torsion is shown to depend only on the Galois group G. For algebraic number fields and function fields, we give a complete classification of those G for which T is nontrivial. Let K/E be a proper extension of infinite fields. Brandis [B] proved in 1965 that K*/E*, the quotient of the multiplicative groups, is never finitely generated; and in 1984 Davis and Maroscia [DM] showed that the quotient group always has infinite torsion-free rank except in the following two situations where K*/E* is obviously torsion: (a) K is an algebraic extension of a finite field, or (b) K is purely inseparable over E. Suppose, now, that K/k is a finite algebraic extension and Ex, ... , E, are proper intermediate fields. For t = 2 it was shown in [W] that K*/E\E\ always has infinite rank unless (a) or (b) holds for one of the E¡. The fact that this result did not appear to generalize to more than two intermediate fields was the starting point for this paper. Assuming now that K/k is a finite Galois extension and that k is not an algebraic extension of a finite field, we examine in detail the structure of the groups K*/E¡ ■ ■■ E*. We determine in (1.4) exactly when this quotient group is torsion; and we show that if k is a "reasonable" field, e.g., an algebraic number field or a function field, then K*/E* ■ ■ ■ E* either is torsion or has a free summand of infinite rank. (See (1.5) and (1.8).) The main results in this paper concern the quotient K*/0(K/k), where Q(K/k) is the compositum of the multiplicative groups of all proper intermediate fields. We will see, for example, that K* = Q(K/k) whenever the Galois group contains S4. We also construct examples, one in characteristic 0 and one in characteristic 2, where the Galois group is C(2) x C(2) and K* = &(K/k). Thus it is possible for K* to be the product of the multiplicative groups of three intermediate fields. We show in §2 that K*/Q(K/k) is torsion if and only if the Galois group of K/k is not a Frobenius complement. In §3 we show that if the group

Proceedings of the American Mathematical Society, 1968

International Mathematics Research Notices

Proceedings - Mathematical Sciences, 2017

We give, in Sections 2 and 3, an english translation of: Classes généralisées invariantes, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: from theory to practice, SMM, Springer-Verlag, 2 nd corrected printing 2005. We recall, in Section 4, some structure theorems for finite Zp[G]-modules (G ≃ Z/p Z) obtained in: Sur les ℓ-classes d'idéaux dans les extensions cycliques relatives de degré premier ℓ, Annales de l'Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a p-class group in a cyclic extension of degree p. In Section 5, we apply this to the study of the structure of relative p-class groups of Abelian extensions of prime to p degree, using the Thaine-Ribet-Mazur-Wiles-Kolyvagin "principal theorem", and the notion of "admissible sets of prime numbers" in a cyclic extension of degree p, from: Sur la structure des groupes de classes relatives, Annales de l'Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the p-ramification theory (as dual form of non-ramification theory) and which have become standard in a p-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject. We shall see the field L given, via Class Field Theory, by some Artin group of K (e.g., the Hilbert class field H + K of K associated with the group of principal ideals, in the narrow sense, any ray class field H + K,m associated with a ray group modulo a modulus m of k, in the narrow sense, or more generally any subfield L of these canonical fields, defining Gal(H + K,m /L) by means of a sub-G-module H of the generalized class group Cℓ + K,m ≃ Gal(H + K,m /K)). We intend to give, from the arithmetic of k and elementary local normic computations in K/k, an explicit formula for #Gal(L/K) G = #(Cℓ + K,m /H) G. This order is the degree, over K, of the maximal subfield of L (denoted L ab) which is Abelian over k.

2009

We consider a Galois extension E/F of characteristic 0 and realization fields of finite abelian subgroups G ⊂ GLn(E) of a given exponent t. We assume that G is stable under the natural operation of the Galois group of E/F . It is proven that under some reasonable restrictions for n any E can be a realization field of G, while if all coefficients of matrices in G are algebraic integers there are only finitely many fields E of realization having a given degree d for prescribed integers n and t or prescribed n and d. Some related results and conjectures are considered.

Following a paper by Athanasios Angelakis and Peter Stevenhagen on the determination of imaginary quadratic fields having the same absolute Abelian Galois group A, we study this property for arbitrary number fields. We show that such a property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p-adic obstructions coming from the global units. By restriction to the p-Sylow subgroups of A, we show that the corresponding study is related to a generalization of the classical notion of p-rational fields. However, we obtain (Thms. 2.8, 3.1) non-trivial information about the structure of the profinite group A, for every number field, by application of results of our book on class field theory.