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).