An irreducible polynomialMathworldPlanetmath fF[x] with coefficientsMathworldPlanetmath in a field F is separable if f factors into distinct linear factors over a splitting fieldMathworldPlanetmath K of f.

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

An algebraic field extension K/F is separable if, for each aK, the minimal polynomial of a over F is separable. When F has characteristic zero, every algebraic extension of F is separable; examples of inseparable extensionsPlanetmathPlanetmath include the quotient field K(u)[t]/(tp-u) over the field K(u) of rational functions in one variable, where K has characteristicPlanetmathPlanetmath p>0.

More generally, an arbitrary field extension K/F is defined to be separable if every finitely generatedMathworldPlanetmathPlanetmath intermediate field extension L/F has a transcendence basis SL 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