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In mathematics, a hyper-finite field is an
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
similar in many ways to
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s. 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