separable SeparablePolynomial 2002-01-05 01:53:45 2005-03-05 04:30:44 Definition separable separable polynomial separable extension \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} An irreducible polynomial $f \in F[x]$ with coefficients in a field $F$ is {\em separable} if $f$ factors into distinct linear factors over a splitting field $K$ of $f$. A polynomial $g$ with coefficients in $F$ is {\em separable} if each irreducible factor of $g$ in $F[x]$ is a separable polynomial. An algebraic field extension $K/F$ is {\em separable} if, for each $a \in K$, the minimal polynomial of $a$ over $F$ is separable. When $F$ has characteristic zero, every algebraic extension of $F$ is separable; examples of inseparable extensions include the quotient field $K(u)[t]/(t^p-u)$ over the field $K(u)$ of rational functions in one variable, where $K$ has characteristic $p > 0$. More generally, an arbitrary field extension $K/F$ is defined to be {\em separable} if every finitely generated intermediate field extension $L/F$ has a transcendence basis $S \subset L$ such that $L$ is a separable algebraic extension of $F(S)$.