SeparablyAlgebraicallyClosedField 2006-06-10 01:06:45 2007-07-03 17:21:03 Definition separably algebraically closed % 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 called \emph{separably algebraically closed} if every separable element of the algebraic closure of $K$ belongs to $K$.\newline In the case when $K$ has characteristic 0, it is separably algebraically closed if and only if it is algebraically closed.\newline If $K$ has positive characteristic $p$, $K$ is separably algebraically closed if and only if its algebraic closure is a purely inseparable extension of $K$.