the compositum of a Galois extension and another extension is Galois

Theorem 1.

Let E/K be a Galois extensionMathworldPlanetmath of fields, let F/K be an arbitrary extensionPlanetmathPlanetmathPlanetmath and assume that E and F are both subfieldsMathworldPlanetmath of some other larger field T. The compositum of E and F is here denoted by EF. Then:

  1. 1.

    EF is a Galois extension of F and E is Galois over EF;

  2. 2.

    Let H=Gal(EF/F). The restrictionPlanetmathPlanetmath map:

    H=Gal(EF/F) Gal(E/EF)
    σ σ|E

    is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, where σ|E denotes the restriction of σ to E.

Remark 1.

Notice, however, that if E/F and F/K are both Galois extensions, the extension E/K need not be Galois. See example of normal extensionMathworldPlanetmath for a counterexample.

Title the compositum of a Galois extension and another extension is Galois
Canonical name TheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois
Date of creation 2013-03-22 15:04:13
Last modified on 2013-03-22 15:04:13
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type TheoremMathworldPlanetmath
Classification msc 12F99
Classification msc 11R32
Related topic FundamentalTheoremOfGaloisTheory
Related topic GaloisExtension
Related topic ExampleOfNormalExtension
Related topic ClassNumberDivisibilityInExtensions
Related topic GaloisGroupOfTheCompositumOfTwoGaloisExtensions
Related topic ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility