mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In field theory, a branch of algebra, a field extension L/k is said to be regular if ''k'' is algebraically closed in ''L'' (i.e., k = \hat k where \hat k is the set of elements in ''L'' algebraic over ''k'') and ''L'' is separable over ''k'', o ...

of ''E'') of smaller

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

than ''F'' can be embedded in ''F''. They were introduced by . Every hyper-finite field is a

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 al ...

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

References

* Field (mathematics) {{Abstract-algebra-stub