# Galois group of the compositum of two Galois extensions

###### Theorem 1.

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

1. 1.

The intersection $E\cap F$ is Galois over $K$.

2. 2.

The compositum $EF$ is Galois over $K$. Moreover, the Galois group $\operatorname{Gal}(EF/K)$ is isomorphic to the subgroup $H$ of the direct product $G=\operatorname{Gal}(E/K)\times\operatorname{Gal}(F/K)$ 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 $E\cap F=K$. Then $EF$ is Galois over $K$ and the Galois group is isomorphic to the direct product:

 $\operatorname{Gal}(EF/K)\cong\operatorname{Gal}(E/K)\times\operatorname{Gal}(F% /K).$
Title Galois group of the compositum of two Galois extensions GaloisGroupOfTheCompositumOfTwoGaloisExtensions 2013-03-22 15:04:22 2013-03-22 15:04:22 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 12F99 msc 11R32 CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois GaloisExtension