Pseudo-finite field: Difference between revisions
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In mathematics, a '''pseudo-finite field''' ''F'' is an infinite model of the [[first-order logic|first-order]] [[theory (mathematical logic)|theory]] of [[finite field]]s. This is equivalent to the condition that ''F'' is [[quasi-finite field|quasi-finite]] (perfect with a unique extension of every positive degree) and [[pseudo algebraically closed]] (every absolutely irreducible variety over ''F'' has a point defined over ''F''). Every [[hyperfinite field]] is pseudo |
In mathematics, a '''pseudo-finite field''' ''F'' is an infinite model of the [[first-order logic|first-order]] [[theory (mathematical logic)|theory]] of [[finite field]]s. This is equivalent to the condition that ''F'' is [[quasi-finite field|quasi-finite]] (perfect with a unique extension of every positive degree) and [[pseudo algebraically closed]] (every absolutely irreducible variety over ''F'' has a point defined over ''F''). Every [[hyperfinite field]] is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal [[ultraproduct]] of finite fields is pseudo-finite. |
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Pseudo-finite fields were introduced by {{harvs|txt|last=Ax|year=1968|authorlink=James Ax}}. |
Pseudo-finite fields were introduced by {{harvs|txt|last=Ax|year=1968|authorlink=James Ax}}. |
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Revision as of 03:45, 13 November 2012
In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudo-finite.
Pseudo-finite fields were introduced by Ax (1968).
References
- Ax, James (1968), "The Elementary Theory of Finite Fields", Annals of Mathematics, Second Series, 88 (2), Annals of Mathematics: 239–271, ISSN 0003-486X, MR 0229613, Zbl 0195.05701
- Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 11 (3rd revised ed.), Springer-Verlag, pp. 448–453, ISBN 978-3-540-77269-9, Zbl 1145.12001