Submission date: 13 February 2015.

Abstract:

We prove the following propositions.
Theorem 1: Let M be a subfield of a fixed algebraic closure \tilde Q of Q whose existential elementary theory is decidable (resp. primitively decidable). Then, M is conjugate to a recursive (resp. primitive recursive) subfield L \subset \tilde Q.
Theorem 2: For each positive integer e there are infinitely many e-tuples σ \in \Gal(Q)^e such that the field \tilde Q( σ) -- the fixed field of σ, is recursive in \tildeQ and its elementary theory is decidable. Moreover, \tilde Q(σ) is PAC and \Gal(\tildeQ(σ))) is isomorphic to the free profinite group on e generators.

Mathematics Subject Classification: 12E30

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Full text arXiv 1502.03885: pdf, ps.