On decidable algebraic fields

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 $\boldsymbol σ\in \Gal(\Q)^e$ such that the field $\tilde \Q( {\boldsymbol σ})$ -- the fixed field of $\boldsymbol σ$, is recursive in $\tilde\Q$ and its elementary theory is decidable. Moreover, $\tilde \Q(\boldsymbol σ)$ is PAC and $\Gal(\tilde\Q(\boldsymbol σ))$ is isomorphic to the free profinite group on $e$ generators.

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