On Galois groups and PAC substructures

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.