We study the automorphism group of the algebraic closure of a substructure A of a pseudofinite 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.

The model companion of the theory of fields with valuation and automorphism exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion.