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.
- Ax, James (1968), "The Elementary Theory of Finite Fields", Annals of Mathematics, Second Series, Annals of Mathematics, 88 (2): 239–271, doi:10.2307/1970573, ISSN 0003-486X, JSTOR 1970573, MR 0229613, Zbl 0195.05701
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