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algebraic closure

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**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 lemmaMcCarthy (1991) p.21M. F. Atiyah and I. G. Macdonald (1969).

*Introduction to commutative algebra*. Addison-Wesley publishing Company. pp. 11â€“12.Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma,{{Citation|first=Bernhard|last=Banaschewski|title=Algebraic closure without choice.|journal=Z. Math. Logik Grundlagen Math.|volume=38|issue=4|pages=383â€“385|year=1992|zbl=0739.03027}}Mathoverflow discussion 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.

## 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**(Ï€). - 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).{{citation | title=Infinite Algebraic Extensions of Finite Fields | volume=95 | series=Contemporary Mathematics | first1=Joel V. | last1=Brawley | first2=George E. | last2=Schnibben | publisher=American Mathematical Society | year=1989 | isbn=978-0-8218-5428-0 | contribution=2.2 The Algebraic Closure of a Finite Field | pages=22â€“23 | url=https://books.google.com/books?id=0HNfpAsMXhUC&pg=PA22 | zbl=0674.12009}}.

## Existence of an algebraic closure and splitting fields

Let S = { f_{lambda} | lambda in Lambda} be the set of all monic irreducible polynomials in*K*[

*x*].For each f_{lambda} in S, introduce new variables u_{lambda,1},ldots,u_{lambda,d} where d = {rm degree}(f_{lambda}).Let

*R*be the polynomial ring over

*K*generated by u_{lambda,i} for all lambda in Lambda and all i leq {rm degree}(f_{lambda}). Write

f_{lambda} - prod_{i=1}^d (x-u_{lambda,i}) = sum_{j=0}^{d-1} r_{lambda,j} cdot x^j in R[x]

with r_{lambda,j} in R.Let *I*be the ideal in

*R*generated by the r_{lambda,j}. 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 f_{lambda} with coefficients in

*K*splits as the product of x-(u_{lambda,i} + M), 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*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*. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of

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

*K*is contained in a

*separably-closed*algebraic extension field. It is unique (up to isomorphism).McCarthy (1991) p.22The 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*, K(X)(sqrt[p]{X}) supset K(X) is a non-separable algebraic field extension.In general, the absolute Galois group of

*K*is the Galois group of

*Ksep*over

*K*.BOOK, Fried, Michael D., Jarden, Moshe, Field arithmetic, 3rd, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 11, Springer-Verlag, 2008, 978-3-540-77269-9, 1145.12001, 12,

## See also

## References

{{reflist}}- BOOK, Irving, Kaplansky, Irving Kaplansky, Fields and rings, Second, Chicago lectures in mathematics, University of Chicago Press, 1972, 0-226-42451-0, 1001.16500,
- BOOK, McCarthy, Paul J., Algebraic extensions of fields, Corrected reprint of the 2nd, 0768.12001, New York, Dover Publications, 1991,

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