Galois group of the compositum of two Galois extensionsGaloisGroupOfTheCompositumOfTwoGaloisExtensions2005-02-22 10:30:072005-03-10 15:39:17Theoremcomposite fieldcompositumGalois group% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Gal}{\operatorname{Gal}}\begin{thm}
Let $E$ and $F$ be Galois extensions of a field $K$. Then:
\begin{enumerate}
\item The intersection $E\cap F$ is Galois over $K$.\\
\item The compositum $EF$ is Galois over $K$. Moreover, the Galois group $\Gal(EF/K)$ is isomorphic to the subgroup $H$ of the direct product $G=\Gal(E/K)\times \Gal(F/K)$ 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.
\end{enumerate}
\end{thm}
\begin{cor}
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:
$$\Gal(EF/K)\cong \Gal(E/K) \times \Gal(F/K).$$
\end{cor}