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A [[Field_(mathematics)|field]] F is '''algebraically closed''' if there is a solution to every [[Polynomial|polynomial]] over F in one variable. Every field is contained in an algebraically closed field and there is a minimal one, called the algebraic closure of F.

A [[Field_(mathematics)|field]] F is '''algebraically closed''' if there is a solution to every [[Polynomial|polynomial]] over F in one variable. Every field is contained in an algebraically closed field and there is a minimal one, called the '''algebraic closure''' of F. This general fact uses [[Zorn's Lemma]], which is equivalent to the [[Axiom of Choice]] (AC). In specific cases AC may sometimes be avoided.

The definition of algebraically closed has several other equivalent forms:

The definition of algebraically closed has several other equivalent forms:

<li> F is a maximal field of its [[Transcendence_degree|transcendence degree]].

<li> F is a maximal field of its [[Transcendence_degree|transcendence degree]].

<li> F is the union of all its [[Finitely-generated|finitely-generated]] [[Localization|localized]] [[Ring_(mathematics)|subrings]].

<li> F is the union of all its [[Finitely-generated|finitely-generated]] [[Localization|localized]] [[Ring_(mathematics)|subrings]].

<li> Every [[Algebraic_variety|algebraic variety]] defined over F has a [[Point_(mathematics)|point]].

<li> Every [[Algebraic_variety|algebraic variety]] defined over F has a [[point]].

<li> F is not finite and the [[Automorphism|automorphism]] group of F has [[Cardinality|cardinality]] strictly greater than the cardinality of F.

<li> F is not finite and the [[Automorphism|automorphism]] group of F has [[Cardinality|cardinality]] strictly greater than the cardinality of F.

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The [[Complex_number|complex numbers]] <math>\mathbb{C}</math> are algebraically closed by the [[Fundamental_Theorem_of_Algebra|Fundamental Theorem of Algebra]], however the [[Rational_number|rational numbers]] <math>\mathbb{Q}</math> are not since x²+1 does not have a root. This same polynomial shows the [[Real_number|real numbers]] <math>\mathbb{R}</math> are also not algebraically closed, yielding as a [[Corollary|corollary]] (by the equivalence) that <math>\textrm{Aut}(\mathbb{R})</math> has [[Cardinality|cardinality]] the [[Continuum|continuum]].

The [[Complex_number|complex numbers]] <math>\mathbb{C}</math> are algebraically closed by the [[Fundamental_Theorem_of_Algebra|Fundamental Theorem of Algebra]], however the [[Rational_number|rational numbers]] <math>\mathbb{Q}</math> are not since x²+1 does not have a root. This same polynomial shows the [[Real_number|real numbers]] <math>\mathbb{R}</math> are also not algebraically closed, yielding as a [[Corollary|corollary]] (by the equivalence) that <math>\textrm{Aut}(\mathbb{R})</math> has [[Cardinality|cardinality]] the [[Continuum|continuum]].

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Algebra]]

[[Category:Algebra]]