splitting field

Let fF[x] be a polynomialMathworldPlanetmathPlanetmathPlanetmath over a field F. A splitting fieldMathworldPlanetmath for f is a field extension K of F such that

  1. 1.

    f splits (factors into a product of linear factors) in K[x],

  2. 2.

    K is the smallest field with this property (any sub-extension field of K which satisfies the first property is equal to K).

Theorem: Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic. A splitting field is always a normal extensionMathworldPlanetmath of the ground field.

Title splitting field
Canonical name SplittingField
Date of creation 2013-03-22 12:08:01
Last modified on 2013-03-22 12:08:01
Owner djao (24)
Last modified by djao (24)
Numerical id 7
Author djao (24)
Entry type Definition
Classification msc 12F05
Related topic NormalExtension