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
Canonical name	Separable
Date of creation	2013-03-22 12:08:04
Last modified on	2013-03-22 12:08:04
Owner	djao (24)
Last modified by	djao (24)
Numerical id	13
Author	djao (24)
Entry type	Definition
Classification	msc 12F10
Classification	msc 11R32
Related topic	PerfectField
Defines	separable
Defines	separable polynomial
Defines	separable extension