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Large normal extension of Hilbertian fields. (English) Zbl 0873.12001
Let $$G(K)$$ denote the absolute Galois group of a field $$K$$. For $$\sigma \in G(K)^e$$ let $$K_s[\sigma]$$ be the largest Galois extension of $$K$$ which is contained in the fix field $$K_s(\sigma)$$ of $$\sigma$$. The author shows that for a countable separably Hilbertian field $$K$$, for almost all $$\sigma\in G(K)^e$$ the field $$K_s[\sigma]$$ has these properties:
(1) $$K_s[\sigma]$$ is PAC (pseudo algebraically closed) (Every non-empty absolutely irreducible variety defined over $$K_s[\sigma]$$ has a $$K_s[\sigma]$$-rational point).
(2) $$K_s[\sigma]$$ is $$\omega$$-free (that means its absolute Galois group is a free profinite group on countably many generators. Hence $$G(K_s[\sigma])\simeq\widehat F_\omega)$$.
By a result of Roquette, $$K_s [\sigma]$$ is separably Hilbertian for these $$\sigma\in G(K)^e$$. Being PAC follows from a recent result of K. Neumann. Since for almost all $$\sigma\in G(K)^e$$ the Galois group of $$K_s [\sigma]$$ is a closed normal subgroup of $$\widetilde F_\omega$$, we can apply a result of Melnikov. It states that $$N=G(K_s [\sigma])$$ and $$\widehat F_\omega$$ are isomorphic if $$N$$ has certain quotients. Sections 3 and 4 contain some group theoretic applications and some consequences for decidability.

##### MSC:
 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 12F99 Field extensions 12L05 Decidability and field theory
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