Geometric representation in the theory of pseudo-finite fields

TL;DR

This paper investigates the automorphism groups of algebraic closures in pseudo-finite fields, establishing that finite groups geometrically represented in such fields are necessarily abelian, thus advancing understanding of their algebraic structure.

Contribution

It proves that any finite group geometrically represented in a pseudo-finite field must be abelian, addressing open questions in the field.

Findings

Finite groups in pseudo-finite fields are abelian.

Automorphism groups of algebraic closures are characterized.

Answers to open questions in the theory of pseudo-finite fields.

Abstract

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.