algebraically closed

A field K is algebraically closedMathworldPlanetmath if every non-constant polynomialPlanetmathPlanetmath in K[X] has a root in K.

An extension fieldMathworldPlanetmath L of K is an algebraic closure of K if L is algebraically closed and every element of L is algebraic over K. Using the axiom of choiceMathworldPlanetmath, one can show that any field has an algebraic closure. Moreover, any two algebraic closures of a field are isomorphic as fields, but not necessarily canonically isomorphic.

Title algebraically closed
Canonical name AlgebraicallyClosed
Date of creation 2013-03-22 12:12:06
Last modified on 2013-03-22 12:12:06
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Definition
Classification msc 12F05
Defines algebraic closure