On algebraic closure in pseudofinite fields. (English) Zbl 1273.03126

Summary: 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.


03C60 Model-theoretic algebra
12L12 Model theory of fields
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