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

TL;DR

This paper constructs a first-order logical statement for any finite group that characterizes the existence of a Galois extension with that group over any field, linking the Inverse Galois Problem to the field's first-order theory.

Contribution

It introduces a uniform first-order statement for finite groups that encodes Galois extension existence, connecting the Inverse Galois Problem to the logical theory of fields.

Findings

Existence of a first-order statement $S(G)$ for each finite group $G$

Decidability of the Inverse Galois Problem reduces to the first-order theory of the field

Effective procedure to produce $S(G)$ from the multiplication table of $G$

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.