# separable

An irreducible polynomial $f\in F[x]$ with coefficients in a field $F$ is separable if $f$ factors into distinct linear factors over a splitting field $K$ of $f$.

A polynomial $g$ with coefficients in $F$ is separable if each irreducible factor of $g$ in $F[x]$ is a separable polynomial.

An algebraic field extension $K/F$ is 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 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 $F(S)$.

Title separable Separable 2013-03-22 12:08:04 2013-03-22 12:08:04 djao (24) djao (24) 13 djao (24) Definition msc 12F10 msc 11R32 PerfectField separable separable polynomial separable extension