PurelyInseparable 2004-11-16 19:13:22 2006-10-18 11:06:54 Definition % 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 \theoremstyle{definition} \newtheorem*{example}{Example} \newcommand{\F}{\mathbb{F}} Let $F$ be a field of characteristic $p > 0$ and let $\alpha$ be an element which is algebraic over $F$. Then $\alpha$ is said to be \emph{purely inseparable} over $F$ if $\alpha^{p^n} \in F$ for some $n \ge 0$. An algebraic field extension $K/F$ is \emph{purely inseparable} if each element of $K$ is purely inseparable over $F$. Purely inseparable extensions have the following property: if $K/F$ is purely inseparable, and $A$ is an algebraic closure of $F$ which contains $K$, then any homomorphism $K \to A$ which fixes $F$ necessarily fixes $K$. Let $K/F$ be an arbitrary algebraic extension. Then there is an intermediate field $E$ such that $K/E$ is purely inseparable, and $E/F$ is separable. \begin{example} Let $s$ be an indeterminate, and let $K = \F_3(s)$ where $\F_3$ is the finite field with $3$ elements. Let $F = \F_3(s^6)$. Then $K/F$ is neither separable, nor purely inseparable. Let $E = \F_3(s^3)$. Then $E/F$ is separable, and $K/E$ is purely inseparable. \end{example}