On decidable algebraic fields

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