PSC Galois extensions of Hilbertian fields

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all σ ∈ G(K)^e, the field K_s[σ] ∩ K_tot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂_p. Then K_p = K_s ∩ K̂_p and K_tot,S = ∩ _p∈S ∩_σ∈G K_p^Σ. G(K) stands for the absolute Galois group of K. For each σ = (σ₁,..., σ_e) ∈ G(K)^e we denote the fixed field of σ₁,..., σ_e in K_s by K_s (σ). The maximal Galois extension of K in K_s (σ) is K_s[σ]. Finally "almost all" means "for all but a set of Haar measure zero".