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).
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