Algebraic extensions of finite corank of hilbertian fields

Abstract

We consider here a hilbertian fieldk and its Galois group (k s/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 fieldk s (σ) of 〈σ〉, …, σe ink s, which containsk. Then the group (k s/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 [k s (σ):k s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk⊆K⊆k s (σ) such that 1<[k s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group (ks/k)e.