In
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 ...
, particularly
abstract algebra, an algebraic closure of a
field ''K'' is an
algebraic extension of ''K'' that is
algebraically closed. It is one of many
closures in mathematics.
Using
Zorn's lemma[McCarthy (1991) p.21][Kaplansky (1972) pp.74-76] or the weaker
ultrafilter lemma, it can be shown that
every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
an
isomorphism that
fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''.
The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''.
To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''.
The algebraic closure of ''K'' is also the smallest algebraically closed field containing ''K'',
because if ''M'' is any algebraically closed field containing ''K'', then the elements of ''M'' that are
algebraic over ''K'' form an algebraic closure of ''K''.
The algebraic closure of a field ''K'' has the same
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 ...
as ''K'' if ''K'' is infinite, and is