Pseudo-finite field
{{more footnotes|date=December 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 {{harvs|txt|last=Ax|year=1968|authorlink=James Ax}}.
References
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- {{Citation | last1=Ax | first1=James | title=The Elementary Theory of Finite Fields | jstor=1970573 | publisher=Annals of Mathematics | series=Second Series | zbl=0195.05701 | mr=0229613 | year=1968 | journal=Annals of Mathematics | issn=0003-486X | volume=88 | issue=2 | pages=239–271 | doi=10.2307/1970573}}
- {{citation | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 | pages=448–453 }}