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 . Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.