On Galois groups and PAC substructures

TL;DR

This paper characterizes Galois groups in stable theories, showing their correspondence with profinite groups and PAC substructures, and explores their properties and realizations within model theory.

Contribution

It establishes a correspondence between profinite groups and Galois groups in stable theories, and characterizes PAC substructures with projective Galois groups.

Findings

Profinite groups are exactly Galois groups of some extensions in stable theories.

PAC substructures have projective absolute Galois groups.

Certain Galois groups are described via the universal Frattini cover.

Abstract

We show that for an arbitrary stable theory T, a group G is profinite if and only if G occurs as a Galois group of some Galois extension inside a monster model of T. We prove that any PAC substructure of the monster model of T has projective absolute Galois group. Moreover, any projective profinite group G is isomorphic to the absolute Galois group of some substructure P of the monster model. If T is omega-stable, then P can be chosen to be PAC. Finally, we provide a description of some Galois groups of existentially closed substructures with G-action in the terms of the universal Frattini cover. Such structures might be understood as a new examples of PAC structures.