Moshe Jarden

Algebraic extensions of finite corank of hilbertian fields

We consider here a hilbertian fieldk and its Galois group (ks/k). For a natural numbere we prove that almost all (σ) ∈(ks/k)e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ1, …, σe is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldks (σ) of 〈σ〉, …, σe inks, which containsk. Then the group (ks/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈ (k/k), τ≠1, of finite order such that [ks (σ):ks (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk⊆K⊆ks (σ) such that 1<[ks (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group(ks/k)e.

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.

Intersections of Local Algebraic Extensions of a Hilbertian Field

The main object of this work is to present a unified generalization of the “free generators theorem” of the author and Geyer’s theorem about the absolute Galois group of the intersection of Henselizations of a countable Hilbertian field K with respect to finitely many absolute values. We equip the absolute Galois group G(K) of K with its Haar measure and use the term “almost all” in the usual sense of measure theory.

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.

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

Pseudo algebraically closed fields over rings

AbstractWe prove that for almost allσ ∈G ℚ the field

On the characterization of local fields by their absolute Galois groups

\tilde{\mathbb{Q}}

l

has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→