On algebraic closure in pseudofinite fields

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 pseudofinite fields, we show that over $A$, algebraic closure agrees with definable closure, as soon as $A$ contains the relative algebraic closure of the prime field.