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Algebraic closure

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 toanisomorphism 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 cardinalityasKifK is infinite, and is countably infiniteifK is finite.^[3]

• 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 fieldofprime power order q, the algebraic closure is a countably infinite field that contains a copy of the field of order qⁿ for each positive integer n (and is in fact the union of these copies).^[6]

Let ${\displaystyle S=\{f_{\lambda }|\lambda \in \Lambda \}}$ be the set of all monic irreducible polynomialsinK[x]. For each ${\displaystyle f_{\lambda }\in S}$, introduce new variables ${\displaystyle u_{\lambda ,1},\ldots ,u_{\lambda ,d}}$ where ${\displaystyle d={\rm {degree}}(f_{\lambda })}$. Let R be the polynomial ring over K generated by ${\displaystyle u_{\lambda ,i}}$ for all ${\displaystyle \lambda \in \Lambda }$ and all ${\displaystyle i\leq {\rm {degree}}(f_{\lambda })}$. Write

${\displaystyle f_{\lambda }-\prod _{i=1}^{d}(x-u_{\lambda ,i})=\sum _{j=0}^{d-1}r_{\lambda ,j}\cdot x^{j}\in R[$x]}

with ${\displaystyle r_{\lambda ,j}\in R}$. Let I be the idealinR generated by the ${\displaystyle r_{\lambda ,j}}$. Since I is strictly smaller than R, Zorn's lemma implies that there exists a maximal ideal MinR that contains I. The field K₁=R/M has the property that every polynomial ${\displaystyle f_{\lambda }}$ with coefficients in K splits as the product of ${\displaystyle x-(u_{\lambda ,i}+M),}$ and hence has all roots in K₁. In the same way, an extension K₂ofK₁ 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 SofK[x], there exists a splitting fieldofS over K.

An algebraic closure K^algofK contains a unique separable extension K^sepofK containing all (algebraic) separable extensions of K within K^alg. This subextension is called a separable closureofK. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of K^sep, of degree > 1. Saying this another way, 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, ${\displaystyle K($X)({\sqrt[{p}]{X}})\supset K(X)} is a non-separable algebraic field extension.

In general, the absolute Galois groupofK is the Galois group of K^sep over K.^[8]

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

• 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.