Algebraic extensions of finite corank of hilbertian fields

We consider here a hilbertian field k and its Galois group[Figure not available: see fulltext.] (k _s/k). For a natural number e we prove that almost all (σ) ∈[Figure not available: see fulltext.](k_s/k)^e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ₁, ..., σ_e is a free pro-finite group with e generators. (2) Let K be a proper subfield of the fixed field k _s (σ) of 〈σ〉, ..., σ_e in k _s, which contains k. Then the group[Figure not available: see fulltext.] (k _s/K) cannot be topologically generated by less then e+1 elements. (3) There does not exist a τ ∈[Figure not available: see fulltext.] (k/k), τ≠1, of finite order such that [k _s (σ):k _s (σ, τ)]<∞. (4) If e=1, there does not exist a field k⊆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[Figure not available: see fulltext.](k_s/k)^e.