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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.

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