Algebraic Closure

In 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, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to 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 which are algebraic over K form an algebraic closure of K.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.

Read more about Algebraic Closure:  Examples, Separable Closure

Other articles related to "algebraic closure, algebraic, closure":

Finite Field - Properties and Facts - Containment
... The direct limit of this system is a field, and is an algebraic closure of Fp (or indeed of Fpn for any n), denoted ... fields, the Frobenius automorphism on the algebraic closure of Fp has infinite order (no iterate of it is the identity function on the whole field), and it does not ... That is, there are automorphisms of the algebraic closure which are not iterates of the pth power map ...
p-adic Number - Properties
... The real numbers have only a single proper algebraic extension, the complex numbers in other words, this quadratic extension is already algebraically closed ... By contrast, the algebraic closure of the p-adic numbers has infinite degree, i.e ... Qp has infinitely many inequivalent algebraic extensions ...
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... However, for each prime p there is an algebraic closure of any finite field of characteristic p, as below ...
Algebraic Closure - Separable Closure
... An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separable extensions of K within Kalg ... This subextension is called a separable closure of K ... Saying this another way, K is contained in a separably-closed algebraic extension field ...
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... Finitary closure operators play a relatively prominent role in universal algebra, and in this context they are traditionally called algebraic closure operators ... This gives rise to a finitary closure operator ... group the subgroup generated by it, and similarly for fields and all other types of algebraic structures ...

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