# 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[1][2][3] or the weaker ultrafilter lemma,[4][5] 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* 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).[6]

## 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*.
The field *K*_{1}=*R*/*M* has the property that every polynomial with coefficients in *K* splits as the product of and hence has all roots in *K*_{1}. In the same way, an extension *K*_{2} of *K*_{1} can be constructed, etc. The union of all these extensions is the algebraic closure of *K*, because any polynomial with coefficients in this new field has its coefficients in some *K*_{n} with sufficiently large *n*, and then its roots are in *K*_{n+1}, and hence in the union itself.

It can be shown along the same lines 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).[7]

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*.[8]

## References

- McCarthy (1991) p.21
- M. F. Atiyah and I. G. Macdonald (1969).
*Introduction to commutative algebra*. Addison-Wesley publishing Company. pp. 11–12. - Kaplansky (1972) pp.74-76
- Banaschewski, Bernhard (1992), "Algebraic closure without choice.",
*Z. Math. Logik Grundlagen Math.*,**38**(4): 383–385, Zbl 0739.03027 - Mathoverflow discussion
- 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.