Galois group of the compositum of two Galois extensions

Theorem 1.

Let E and F be Galois extensionsMathworldPlanetmath of a field K. Then:

  1. 1.

    The intersectionDlmfMathworldPlanetmath EF is Galois over K.

  2. 2.

    The compositum EF is Galois over K. Moreover, the Galois groupMathworldPlanetmath Gal(EF/K) is isomorphicPlanetmathPlanetmathPlanetmath to the subgroupMathworldPlanetmathPlanetmath H of the direct productPlanetmathPlanetmathPlanetmathPlanetmath G=Gal(E/K)×Gal(F/K) given by:


    i. e. H consists of pairs of elements of G whose restrictionsPlanetmathPlanetmathPlanetmath to EF are equal.

Corollary 1.

Let E and F be Galois extensions of a field K such that EF=K. Then EF is Galois over K and the Galois group is isomorphic to the direct product:

Title Galois group of the compositum of two Galois extensions
Canonical name GaloisGroupOfTheCompositumOfTwoGaloisExtensions
Date of creation 2013-03-22 15:04:22
Last modified on 2013-03-22 15:04:22
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 12F99
Classification msc 11R32
Related topic CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois
Related topic GaloisExtension