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 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.

9pp, to appear in JSL. Several corrections in new version; Remark (ii) replaced by Example 11 and Remark 12. Thanks to referee for two very useful reports