Dan Haran

Regular split embedding problems over complete valued fields

Abstract We give an elementary self-contained proof of the following result, which Pop proved with methods of rigid geometry.

Galois groups over complete valued fields

We propose an elementary algebraic approach to the patching of Galois groups. We prove that every finite group is regularly realizable over the field of rational functions in one variable over a complete discrete valued field.

Galois stratification over Frobenius fields

Let K be an infinite field finitely generated over its prime lield. Denote by G(K) = .‘a(K,/K) the absolute Galois group of K. The set G(K)‘, for e a positive integer, is equipped with the normalized Haar measure, ,U =,uu,, induced from the measure of G(K) that assigns to G(L) the value l/[L: K], if L/K is a finite separable extension. If o = (a, ,..., a,) E G(K)‘, then we denote by g(a) the fixed field of u,,..., ue in 2 (= the algebraic closure of K). Denote also by Y(K) the first-order language of fields enriched with constant symbols for the elements of K. For every sentence 0 of Y(K) we define A,(B) = {a E G(K)=Il?(u) i= 19). Further we denote by T,(K) the theory of all sentences 19 of Y(K) with ,u(A,(O)) = 1. In [ 13, Theorem 7.3 1 the following is shown.

On Galois groups over pythagorean and semi-real closed fields

We call a fieldK semi-real closed if it is algebraically maximal with respect to a semi-ordering. It is proved that (as in the case of real closed fields) this is a Galois-theoretic property. We give a recursive description of all absolute Galois groups of semi-real closed fields of finite rank.

EFFECTIVE COUNTING OF THE POINTS OF DEFINABLE SETS OVER FINITE FIELDS

Given a formula in the language of fields we use Galois stratification to establish an effective algorithm to estimate the number of points over finite fields that satisfy the formula

Embedding covers and the theory of frobenius fields

We show that the theory of Frobenius fields is decidable. This is conjectured in [4], [8] and [13], and we prove it by solving a group theoretic problem to which this question is reduced there. To do this we present and develop the notion of embedding covers of finite and pro-finite groups. We also answer two other problems from [8], again by solving a corresponding group theoretic problem: A finite extension of a Frobenius field need not be Frobenius and there are PAC fields which are not Frobenius fields.

Permanence criteria for semi-free profinite groups

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x, y)) have free absolute Galois groups.

NORMAL SUBGROUPS OF PROFINITE GROUPS OF FINITE COHOMOLOGICAL DIMENSION

We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincare group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N). Some other cases when G/N has virtually finite cohomological dimension are considered too. If G is profinite, the case of N projective or the profinite completion of the fundamental group of a compact surface is considered.

Frobenius subgroups of free profinite products

We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of certain type do occur as closed subgroups of free profinite products of two profinite groups. This also solves a question of Pop about prosolvable subgroups of free profinite products.

Maximal abelian subgroups of free profinite groups

THEOREM. Let F be the free profinite group on a set X, where \X\ > 2, and let n be a non-empty set of primes. Then F has a maximal abelian subgroup isomorphic to HpEn Zp. The idea of the proof is the following: we show that A — Ylpe7I1p is a free factor of Pa, i.e. fia ̂ A *B for some profinite group B. To conclude from this that A is a maximal abelian subgroup of Fa (the general case then follows from this one), we show that CA*B(a) = CA(a) (*)