Beyarslan, Özlem; Hrushovski, Ehud On algebraic closure in pseudofinite fields. (English) Zbl 1273.03126 J. Symb. Log. 77, No. 4, 1057-1066 (2012). 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. Cited in 1 ReviewCited in 2 Documents MSC: 03C60 Model-theoretic algebra 12L12 Model theory of fields Keywords:automorphism group; algebraic closure; pseudo-finite field; roots of unity; definable closure PDF BibTeX XML Cite \textit{Ö. Beyarslan} and \textit{E. Hrushovski}, J. Symb. Log. 77, No. 4, 1057--1066 (2012; Zbl 1273.03126) Full Text: DOI arXiv Euclid OpenURL References: [1] Field arithmetic 11 (2005) [2] DOI: 10.1090/S0002-9947-99-02498-8 · Zbl 0922.03054 [3] DOI: 10.2307/1970573 · Zbl 0195.05701 [4] Arithmetic algebraic geometry (Park City, UT, 1999) (2009) [5] Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften 116 pp 601– (1907) [6] DOI: 10.1090/S0002-9939-01-06001-4 · Zbl 1012.12007 [7] Model theory and applications 11 pp 151– (2002) [8] Finitely axiomatizable 1 categorical theories 59 pp 838– (1994) [9] Local fields (1979) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.