Some definable Galois theory and examples

TL;DR

This paper explores a definable Galois theory framework, clarifies the Galois correspondence for definable automorphism groups, and relates it to differential equations with non-closed constants, also refining generalized strongly normal extensions.

Contribution

It explicitly develops the Galois correspondence in the context of definable automorphism groups and improves the theory of generalized strongly normal extensions.

Findings

Clarified the Galois correspondence for definable automorphism groups

Connected Galois theory to algebraic differential equations with non-closed constants

Enhanced definitions and results on generalized strongly normal extensions

Abstract

We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not "closed" in suitable senses. We also improve the definitions and results on generalized strongly normal extensions.