On Galois groups and PAC substructures
Volume 250 / 2020
Fundamenta Mathematicae 250 (2020), 151-177
MSC: Primary 03C95; Secondary 03C45, 03C07.
DOI: 10.4064/fm788-10-2019
Published online: 31 January 2020
Abstract
(1) We show that for an arbitrary stable theory $T$, a group $G$ is profinite if and only if $G$ occurs as the Galois group of some Galois extension inside a monster model of $T$.
(2) We prove that any PAC substructure of the monster model of $T$ has projective absolute Galois group.
(3) Moreover, any projective profinite group $G$ is isomorphic to the absolute Galois group of a definably closed substructure $P$ of the monster model. If $T$ is $\omega $-stable, then $P$ can be chosen to be PAC.
(4) Finally, we provide a description of some Galois groups of existentially closed substructures with $G$-action in terms of the universal Frattini cover. Such structures might be understood as a new source of examples of PAC structures.
