Existentially closed fields with finite group actions

TL;DR

This paper investigates the algebraic and model-theoretic properties of existentially closed fields equipped with a finite group action, revealing their pseudo-algebraic closure properties and situating them within the broader context of fields with group scheme actions.

Contribution

It introduces a detailed study of existentially closed fields with finite group actions, highlighting their pseudo-algebraic closure and expanding the model theory of fields with group scheme actions.

Findings

Existentially closed fields with finite group actions are pseudo-algebraically closed.

These fields exhibit strong algebraic closure properties.

The work extends the model theory of fields to include finite group scheme actions.

Abstract

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.