AlgebraicallyClosed 2002-01-21 21:56:04 2008-09-13 16:31:07 Definition algebraic closure % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A field $K$ is \emph{algebraically closed} if every non-constant polynomial in $K[X]$ has a root in $K$. An extension field $L$ of $K$ is an \emph{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.