# 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,^{[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]}

## Existence of an algebraic closure and splitting fields

Let be the set of all monic irreducible polynomials in *K*[*x*].
For each , introduce new variables where .
Let *R* be the polynomial ring over *K* generated by for all and all . Write

with .
Let *I* be the ideal in *R* generated by the . Since *I* is strictly smaller than *R*,
Zorn's lemma implies that there exists a maximal ideal *M* in *R* that contains *I*.
Now the field *R*/*M* is an algebraic closure of *K*: every polynomial splits as the
product of the .

The same proof also shows that for any subset *S* of *K*[*x*], there exists a splitting field of *S* over *K*.

## 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 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. - 1 2 Kaplansky (1972) pp.74-76
- ↑ Brawley, Joel V.; Schnibben, George E. (1989), "2.2 The Algebraic Closure of a Finite Field",
*Infinite Algebraic Extensions of Finite Fields*, Contemporary Mathematics,**95**, American Mathematical Society, pp. 22–23, ISBN 978-0-8218-5428-0, Zbl 0674.12009. - ↑ McCarthy (1991) p.22
- ↑ Fried, Michael D.; Jarden, Moshe (2008).
*Field arithmetic*. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.**11**(3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.

- Kaplansky, Irving (1972).
*Fields and rings*. Chicago lectures in mathematics (Second ed.). University of Chicago Press. ISBN 0-226-42451-0. Zbl 1001.16500. - McCarthy, Paul J. (1991).
*Algebraic extensions of fields*(Corrected reprint of the 2nd ed.). New York: Dover Publications. Zbl 0768.12001.