An invitation to model-theoretic Galois theory

TL;DR

This paper generalizes Galois theory within the framework of model theory, replacing fields with definably closed sets in arbitrary first-order theories, and establishes a duality similar to classical Galois correspondence.

Contribution

It extends Galois theory to arbitrary first-order theories using model-theoretic concepts, providing a new perspective and a simplified exposition of Poizat's work.

Findings

Established Galois duality in model-theoretic setting

Replaced algebraically closed fields with definably closed sets

Provided an accessible exposition requiring minimal background

Abstract

We carry out some of Galois's work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions. This exposition of a special case of Poizat's "Une th\'eorie de Galois imaginaire." (1983) has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.