First-order theory of a field and its Inverse Galois Problem

Abstract

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois extension with the Galois group isomorphic to $G$. Further, there is an effective procedure which takes the table of multiplication of $G$ as its input and produces $S_G$. Therefore, given a field $K$, the Inverse Galois Problem for $K$, that is, the problem of deciding whether $K$ has a Galois extension with a particular Galois group as input, is Turing reducible to the first-order theory of $K$. Similar results hold for the Finite Split Embedding Problem and the Inverse Automorphism Problem.