On decidable algebraic fields

TL;DR

This paper establishes conditions under which algebraic subfields of the algebraic closure of rationals are decidable and recursive, and constructs infinitely many such fields with specific Galois group properties.

Contribution

It proves that subfields with decidable elementary theories are conjugate to recursive subfields and constructs infinitely many fields with prescribed Galois groups and decidability.

Findings

Subfields with decidable elementary theories are conjugate to recursive subfields.

Existence of infinitely many recursive fields with free profinite Galois groups.

Construction of fields that are PAC with decidable elementary theories.

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