Model theory of fields with finite group scheme actions

TL;DR

This paper develops a model theory for fields with finite group scheme actions, establishing a model companion and applying it to actions of finite groups on fields with finite imperfection degree.

Contribution

It introduces a new model complete theory for fields with finite group actions, generalizing previous results on derivations and Galois actions.

Findings

Proves existence and simplicity of the model companion.

Generalizes previous results on derivations and Galois actions.

Provides a new model complete theory for finite group actions on fields.

Abstract

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse-Schmidt derivations and about Galois actions. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.