In mathematics, a **hyper-finite field** is an uncountable field similar in many ways to finite fields. More precisely a field *F* is called hyper-finite if it is uncountable and quasi-finite, and for every subfield *E*, every absolutely entire *E*-algebra (regular field extension of *E*) of smaller cardinality than *F* can be embedded in *F*. They were introduced by Ax (1968). Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.