Difference between revisions of "Algebraic closure"
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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 x<sup>2</sup>+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 x<sup>2</sup>+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]]. | ||
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Revision as of 16:03, January 12, 2008
A field F is algebraically closed if there is a solution to every 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.
The definition of algebraically closed has several other equivalent forms:
- For every
and every polynomial f(x), there is a 
such that 
.
- The only irreducible polynomials in F[x] have degree 1.
- F has no finite extensions other than itself.
- F is a maximal field of its transcendence degree.
- F is the union of all its finitely-generated localized subrings.
- Every algebraic variety defined over F has a point.
- F is not finite and the automorphism group of F has cardinality strictly greater than the cardinality of F.
The complex numbers 
are algebraically closed by the Fundamental Theorem of Algebra, however the rational numbers 
are not since x2+1 does not have a root. This same polynomial shows the real numbers 
are also not algebraically closed, yielding as a corollary (by the equivalence) that 
has cardinality the continuum.
