On algebraic closure in pseudofinite fields

TL;DR

This paper investigates the automorphism group of algebraic closures in pseudo-finite fields, revealing its dependence on roots of unity and establishing conditions under which algebraic and definable closures coincide.

Contribution

It provides new insights into the structure of algebraic closures in pseudo-finite fields and clarifies when algebraic closure aligns with definable closure.

Findings

Automorphism group behavior depends on roots of unity in the field.

Algebraic closure equals definable closure in most cases.

Alignment occurs when A contains the relative algebraic closure of the prime field.

Abstract

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all completions of the theory of pseudo-finite fields we show that algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.