Galois group of the compositum of two Galois extensions
Theorem 1.
Let $E$ and $F$ be Galois extensions^{} of a field $K$. Then:

1.
The intersection^{} $E\cap F$ is Galois over $K$.

2.
The compositum $EF$ is Galois over $K$. Moreover, the Galois group^{} $\mathrm{Gal}(EF/K)$ is isomorphic^{} to the subgroup^{} $H$ of the direct product^{} $G=\mathrm{Gal}(E/K)\times \mathrm{Gal}(F/K)$ given by:
$$H=\{(\sigma ,\psi ):{\sigma }_{E\cap F}={\psi }_{E\cap F}\}$$ i. e. $H$ consists of pairs of elements of $G$ whose restrictions^{} to $E\cap F$ are equal.
Corollary 1.
Let $E$ and $F$ be Galois extensions of a field $K$ such that $E\mathrm{\cap}F\mathrm{=}K$. Then $E\mathit{}F$ is Galois over $K$ and the Galois group is isomorphic to the direct product:
$$\mathrm{Gal}(EF/K)\cong \mathrm{Gal}(E/K)\times \mathrm{Gal}(F/K).$$ 
Title  Galois group of the compositum of two Galois extensions 

Canonical name  GaloisGroupOfTheCompositumOfTwoGaloisExtensions 
Date of creation  20130322 15:04:22 
Last modified on  20130322 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 