P-Adically Projective Groups as Absolute Galois Groups (original) (raw)
The Absolute Galois Group of the Field of Totally S-Adic Numbers
Nagoya Mathematical Journal, 2009
For a finite set S of primes of a number field K and for σ1,…, σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1,…,σe in Ktot,S by Ktot,S(σ). We prove that foralmost all σ ∈ Gal(K)e the absolute Galois group of Ktot,S(σ) is the free product of and a free product of local factors over S.
Functiones et Approximatio Commentarii Mathematici, 2012
For a finite set S of local primes of a countable Hilbertian field K and for σ 1 ,. .. , σe ∈ Gal(K) we denote the field of totally S-adic numbers by K tot,S , the fixed field of σ 1 ,. .. , σe in K tot,S by K tot,S (σ), and the maximal Galois extension of K in K tot,S (σ) by K tot,S [σ]. We prove that for almost all σ ∈ Gal(K) e the absolute Galois group of K tot,S [σ] is isomorphic to the free product ofFω and a free product of local factors over S.
3 on the Structure of the Galois Group of the Abelian Closure of a Number Field
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.
On the structure of the Galois group of the Abelian closure of a number field
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.
Detecting pro-pgroups that are not absolute Galois groups, expanded version. ArXiv: math.NT/06xxxxx
2012
Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity. In this paper we present several constraints on such GF, using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF. In the case p> 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 3 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of GF. Finally, in sections 4–6 we construct new families of pro-p-groups which are not absolute Galois groups over any field F. As a consequence of our results, we prove the following limitations on relator shapes of pro-p absolute Galois ...
Absolute Galois groups viewed from small quotients and the Bloch–Kato conjecture
New topological contexts for Galois theory and algebraic geometry (BIRS 2008), 2009
In this paper we concentrate on the relations between the structure of small Galois groups, arithmetic of fields, Bloch-Kato conjecture, and Galois groups of maximal prop quotients of absolute Galois groups. Contents 1. Introduction 1 2. Some work ofŠafarevič 2 3. The Bloch-Kato conjecture 3 4. Classical Hilbert 90 and absolute Galois groups 4 5. Higher Galois cohomology and the Bloch-Kato conjecture. 7 6. Galois theoretic connections 8 Acknowledgements 11 References 11
Galois groups and the multiplicative structure of field extensions
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