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 greater than or equal to 0 an integer. Then, for almost all sigma is an element of G(K)(e), the field K-s [sigma] boolean AND K-tot,K-S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, (K) over bar (p). Then K-p = K-s boolean AND (K) over bar (p) and K-tot,K-S = boolean AND(pis an element ofS) boolean AND(sigmais an element ofG(K)) K-p(sigma). G(K) stands for the absolute Galois group of K. For each sigma = (sigma(1),...,sigma(e)) is an element of G(K)(e) we denote the fixed field of sigma(1),...,sigma(e) in K-s by K-s(sigma). The maximal Galois extension of K in K-s(sigma) is K-s[sigma]. Finally "almost all" means "for all but a set of Haar measure zero".