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.
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Pseudo - finite fields were introduced by Ax ( 1968 ) .
( https://en. wikipedia. org/wiki/Pseudo - finite_field )
