Pseudo-finite field

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 (Annals of Mathematics) 88 (2): 239–271, doi:10.2307/1970573, ISSN 0003-486X, MR 0229613, Zbl 0195.05701
• Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.), Springer-Verlag, pp. 448–453, ISBN 978-3-540-77269-9, Zbl 1145.12001