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 theory of finite fields. This is equivalent to the condition that 'F'' is a [[quasi-finite field]] such that for every finitely generated absolutely entire ''F''-algebra ''R'' there is an ''F''-homomorphism from ''R'' to ''F'', and to the condition that ''F'' is a quasi-finite field such that 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. |
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 a [[quasi-finite field]] such that for every finitely generated absolutely entire ''F''-algebra ''R'' there is an ''F''-homomorphism from ''R'' to ''F'', and to the condition that ''F'' is a quasi-finite field such that 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}}. |
Pseudo-finite fields were introduced by {{harvs|txt|last=Ax|year=1968|authorlink=James Ax}}. |
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==References== |
==References== |
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Revision as of 00:43, 12 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 a quasi-finite field such that for every finitely generated absolutely entire F-algebra R there is an F-homomorphism from R to F, and to the condition that F is a quasi-finite field such that 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