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 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 James Ax in 1968.
Notes
- Ax, James (1968), "The elementary theory of finite fields", Annals of Mathematics, 88 (2): 239–271, doi:, ISSN, JSTOR, MR, Zbl
- Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol.11 (3rd reviseded.), Springer-Verlag, pp.448–453, ISBN978-3-540-77269-9, Zbl