Approximating absolute Galois groups (original) (raw)
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Subgroups of free profinite groups and large subfields of \mathop Q\limits^ \sim
Israel Journal of Mathematics, 1981
We prove that many subgroups of free profinite groups are free, and use this to give new examples of pseudo-algebraically closed subfields of 0 satisfying Hilbert's Irreducibility Theorem, and to solve problems posed by M. Jarden and A. Macintyre. We also find a subfield of I) which does not satisfy Hilbert's Irreducibility Theorem, but all of whose proper finite extensions do.
Introduction to Profinite Groups
2012
A profinite space / group is the projective limit of finite sets / groups. Galois theory offers a natural frame in order to describe Galois groups as profinite groups. Profinite groups have properties that correspond to some of finite groups: e.g., each profinite group does have p-Sylow subgroups for any prime p. In the same vein, every pro-solvable group (the projective limit of an inverse system of finite solvable groups) has Hall subgroups for any given set of primes. Any group can be equipped with the profinite topology turning it into a topological group. A basis of neighbourhoods of the identity-element consists of all normal subgroups of finite index. Any such group allows a completion w.r.t. this topology – the profinite completion. A free profinite group is the profinite completion of a free group. This can be considered an instance of the amalgamated free product and of the HNN extension (Higman-NeumannNeumann). I do not include cohomological topics in this note.
On some finiteness properties of algebraic groups over finitely generated fields
Comptes Rendus Mathematique, 2016
We present several finiteness results for absolutely almost simple algebraic groups over finitely generated fields that are more general than global fields. We also discuss the relations between the various finiteness properties involved in these results, such as the properness of the global-to-local map in the Galois cohomology of a given K-group G relative to a certain natural set V of discrete valuations of K, and the finiteness of the number of isomorphism classes of K-forms of G having, on the one hand, smooth reduction at V and, on the hand, the same isomorphism classes of maximal K-tori as G. Résumé. Nous présentons plusieurs résultats de finitude pour les groupes algébriques absolument presque simples définis sur des corps de type fini plus généraux que les corps globaux. Nous discutons aussi des liens entre les propriétés de finitude divers qui entrent dans le cadre de notre analyse, tels que la propreté de l'application globale-locale dans la cohomologie galoisienne d'un K-groupe G par rapportà un ensemble convenable V de valuations discrètes de K, et la finitude du nombre de K-formes de G ayant, d'une part, bonne réduction en V , et, d'autre part, possédant les même classes d'isomorphisme de K-tores maximaux que G. même ensemble marcheégalement pour les groupes de type G 2. Dans les sections 2 et 3, il s'agit de l'analyse des groupes algébriques absolument presque simples possédant les mêmes classes d'isomorphisme de tores maximaux sur le corps de définition. Plus précisément, si G est un groupe algébrique simplement connexe absolument presque simple défini sur 1
GROUPS DEFINABLE IN SEPARABLY CLOSED FIELDS
We consider the groups which are innitely denable in separably closed elds of nite degree of imperfection. We prove in particular that no new denable groups arise in this way: we show that any group denable in such a eld L is denably isomorphic to the group of L-rational points of an algebraic group dened over L.
On the descending central sequence of absolute Galois groups
arXiv (Cornell University), 2008
Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group G F of F. Namely, the third subgroup G (3) F in the descending p-central sequence of G F is the intersection of all open normal subgroups N such that G F /N is 1, Z/p 2 , or the extra-special group M p 3 of order p 3 and exponent p 2. Determining the profinite groups which are realizable as absolute Galois groups of fields is a major open problem in Galois theory. Our Main Theorem appears to be simple yet powerful restriction on the possible structure of such groups, and on their quotients G F /G
On Torsion-Free Abelian k-Groups
Proceedings of the American Mathematical Society, 1987
It is shown that a knice subgroup with cardinality Ni, of a torsion-free completely decomposable abelian group, is again completely decomposable. Any torsion-free abelian fc-group of cardinality H" has balanced projective dimension < n. Introduction. Recently, Hill and Megibben introduced the concept of a knice subgroup in their study of abelian fc-groups [6] and also while considering the local Warfield groups in [5]. In this paper, we introduce a modified definition of a knice subgroup of a torsion-free abelian group. This helps us to extend the results of Hill and Megibben [6] and also simplify the proofs of their main theorems. Specifically we show that a knice subgroup with cardinality < Ni, of a torsion-free completely decomposable abelian group, is again completely decomposable. This enables us to prove that any torsion-free abelian fc-group (in particular, a separable group) of cardinality < Nn has balanced projective dimension < n. All the groups that we consider here are torsion-free and abelian. We generally follow the notation and terminology of L. Fuchs [3]. Let P denote the set of all primes. By a height sequence we mean a sequence s = (sp), p G P, where each 5p is a nonnegative integer or the symbol oo. If G is a torsion-free group and x G G, then |x| denotes the height sequence of x where, for each p G P, \x\p denotes the height of x at the prime p. For any height sequence s = (sp), ps is the height sequence (tp), where tp-sp + 1 and tq = sq for all q / p. G(s) denotes the subgroup {x G G: \x\> s}. G(s*) is the subgroup generated by the set {x G G(s): J2pepi\x\p ~ sp) 's unbounded}. Two height sequences (sp) and (tp) are said to be equivalent if YlPep \sp ~ ¿p\ 18 finite.
Proceedings of the American Mathematical Society, 2007
Recently, it has been shown by Harbater and Stevenson that a profinite group G G is free profinite of infinite rank m m if and only if G G is projective and m m -quasifree. The latter condition requires the existence of m m distinct solutions to certain embedding problems for G G . In this paper we provide several new non-trivial examples of m m -quasifree groups, projective and non-projective. Our main result is that open subgroups of m m -quasifree groups are m m -quasifree.
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.
P-Adically Projective Groups as Absolute Galois Groups
2006
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 for almost all σ ∈ Gal(K) e the absolute Galois group of Ktot,S(σ) is the free product ofFe and a free product of local factors over S.