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Pseudo-finite field

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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.

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
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