# Algebraic closure

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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,^{[1]}^{[2]}^{[3]} 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* that 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.^{[3]}

## Examples

- The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

- The algebraic closure of the field of rational numbers is the field of algebraic numbers.

- There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of
**Q**(π).

- For a finite field of prime power order
*q*, the algebraic closure is a countably infinite field that contains a copy of the field of order*q*^{n}for each positive integer*n*(and is in fact the union of these copies).^{[4]}

## Separable closure

An algebraic closure *K ^{alg}* of

*K*contains a unique separable extension

*K*of

^{sep}*K*containing all (algebraic) separable extensions of

*K*within

*K*. This subextension is called a

^{alg}**separable closure**of

*K*. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of

*K*, of degree > 1. Saying this another way,

^{sep}*K*is contained in a

*separably-closed*algebraic extension field. It is essentially unique (up to isomorphism).

^{[5]}

The separable closure is the full algebraic closure if and only if *K* is a perfect field. For example, if *K* is a field of characteristic *p* and if *X* is transcendental over *K*, is a non-separable algebraic field extension.

In general, the absolute Galois group of *K* is the Galois group of *K ^{sep}* over

*K*.

^{[6]}

## See also

## References

- ↑ McCarthy (1991) p.21
- ↑ M. F. Atiyah and I. G. Macdonald (1969).
*Introduction to commutative algebra*. Addison-Wesley publishing Company. pp. 11-12. - ↑
^{3.0}^{3.1}Kaplansky (1972) pp.74-76 - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ McCarthy (1991) p.22
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