Algebraic closure

Inmathematics,particularlyabstract algebra,analgebraic closureof afieldKis analgebraic extensionofKthat isalgebraically closed.It is one of manyclosuresin mathematics.

UsingZorn's lemmaor the weakerultrafilter lemma,it can be shown thatevery field has an algebraic closure,and that the algebraic closure of a fieldKis uniqueup toanisomorphismthatfixesevery member ofK.Because of this essential uniqueness, we often speak ofthealgebraic closure ofK,rather thananalgebraic closure ofK.

The algebraic closure of a fieldKcan be thought of as the largest algebraic extension ofK. To see this, note that ifLis any algebraic extension ofK,then the algebraic closure ofLis also an algebraic closure ofK,and soLis contained within the algebraic closure ofK. The algebraic closure ofKis also the smallest algebraically closed field containingK, because ifMis any algebraically closed field containingK,then the elements ofMthat arealgebraic overKform an algebraic closure ofK.

The algebraic closure of a fieldKhas the samecardinalityasKifKis infinite, and iscountably infiniteifKis finite.

Examples

• Thefundamental theorem of algebrastates that the algebraic closure of the field ofreal numbersis the field ofcomplex numbers.
• The algebraic closure of the field ofrational numbersis the field ofalgebraic 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 oftranscendental extensionsof the rational numbers, e.g. the algebraic closure ofQ(π).
• For afinite fieldofprimepower orderq,the algebraic closure is acountably infinitefield that contains a copy of the field of orderqⁿfor each positiveintegern(and is in fact the union of these copies).

Existence of an algebraic closure and splitting fields

Let$S=\{f_{\lambda }|\lambda \in \Lambda \}$be the set of allmonicirreducible polynomialsinK[x]. For each$f_{\lambda }\in S$,introduce new variables$u_{\lambda,1},\ldots,u_{\lambda,d}$where$d={\rm {degree}}(f_{\lambda })$. LetRbe the polynomial ring overKgenerated 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$r_{\lambda,j}\in R$. LetIbe theidealinRgenerated by the$r_{\lambda,j}$.SinceIis strictly smaller thanR, Zorn's lemma implies that there exists a maximal idealMinRthat containsI. The fieldK₁=R/Mhas the property that every polynomial$f_{\lambda }$with coefficients inKsplits as the product of$x-(u_{\lambda,i}+M),$and hence has all roots inK₁.In the same way, an extensionK₂ofK₁can be constructed, etc. The union of all these extensions is the algebraic closure ofK,because any polynomial with coefficients in this new field has its coefficients in someK_nwith sufficiently largen,and then its roots are inK_n+1,and hence in the union itself.

It can be shown along the same lines that for any subsetSofK[x], there exists asplitting fieldofSoverK.

Separable closure

An algebraic closureK^algofKcontains a uniqueseparable extensionK^sepofKcontaining all (algebraic) separable extensions ofKwithinK^alg.This subextension is called aseparable closureofK.Since a separable extension of a separable extension is again separable, there are no finite separable extensions ofK^sep,of degree > 1. Saying this another way,Kis contained in aseparably-closedalgebraic extension field. It is unique (up toisomorphism).

The separable closure is the full algebraic closure if and only ifKis aperfect field.For example, ifKis a field ofcharacteristicpand ifXis transcendental overK,$K(X)({\sqrt[{p}]{X}})\supset K(X)$is a non-separable algebraic field extension.

In general, theabsolute Galois groupofKis the Galois group ofK^sepoverK.

See also

• Algebraically closed field
• Algebraic extension
• Puiseux expansion
• Complete field

References

• Kaplansky, Irving(1972).Fields and rings.Chicago lectures in mathematics (Second ed.). University of Chicago Press.ISBN0-226-42451-0.Zbl1001.16500.
• McCarthy, Paul J. (1991).Algebraic extensions of fields(Corrected reprint of the 2nd ed.). New York: Dover Publications.Zbl0768.12001.