# 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.

## Key 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.