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On algebraic closure in pseudofinite fields

Published online by Cambridge University Press:  12 March 2014

Özlem Beyarslan  and
Ehud Hrushovski
Özlem Beyarslan
Affiliation:
Department of Mathematics, Boǧaziçi University, 34342 Bebek, Ístanbul, Turkey, E-mail: [email protected]
Ehud Hrushovski
Affiliation:
Department of Mathematics, Hebrew University at Jerusalem, 91904 Jerusalem, Israel, E-mail: [email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 4 , December 2012 , pp. 1057 - 1066
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1]Ax, J., The elementary theory of finite fields, Annals of Mathematics. Second Series, vol. 88 (1968), pp. 239–271.CrossRefGoogle Scholar
[2]Chatzidakis, Z. and Hrushovski, E., Model theory of differencefields, Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 2997–3071.CrossRefGoogle Scholar
[3]Fried, M. and Jarden, M., Field arithmetic, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 11, Springer-Verlag, Berlin, 2005.CrossRefGoogle Scholar
[4]Hahn, H., Über die nichtarchimedischen Grössensysteme, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, vol. 116 (1907), pp. 601–655, (reprinted in: H. Hahn, Gesammelte Abhandlungen I, Springer-Verlag, 1995).Google Scholar
[5]Hrushovski, E., Finitely axiomatizable ℵ1 categorical theories, this Journal, vol. 59 (1994), pp. 838–844.Google Scholar
[6]Hrushovski, E., Pseudo-finite fields and related structures, Model theory and applications, Quaderni di Matematica, vol. 11, Aracne, Rome, 2002, pp. 151–212.Google Scholar
[7]Kedlaya, K. S., The algebraic closure of the power series field in positive characteristic, Proceedings of the American Mathematical Society, vol. 129 (2001), no. 12, pp. 3461–3470.CrossRefGoogle Scholar
[8]Serre, J.-P., Local fields, Springer-Verlag, New York, 1979.CrossRefGoogle Scholar
[9]Tate, J., Galois cohomology, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Mathematical Series, cited 2009, August 25. Available from: http://modular.math.Washington.edu/Tables/Motes/tate-pcmi.html.Google Scholar