algebraically closed

A field $K$ is algebraically closed if every non-constant polynomial in $K[X]$ has a root in $K$.

An extension field $L$ 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 choice, 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