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单词 Separable
释义

separable


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).

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更新时间:2025/5/4 8:52:17