On decidable algebraic fields

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