the compositum of a Galois extension and another extension is Galois

Theorem 1.

Let $E\mathrm{/}K$ be a Galois extension of fields, let $F\mathrm{/}K$ be an arbitrary extension and assume that $E$ and $F$ are both subfields of some other larger field $T$. The compositum of $E$ and $F$ is here denoted by $E\mathit{}F$. Then:

1. $EF$ is a Galois extension of $F$ and $E$ is Galois over $E\cap F$;
   

2. Let $H=\mathrm{Gal}(EF/F)$. The restriction map:

   $H=\mathrm{Gal}(EF/F)$$⟶$$\mathrm{Gal}(E/E\cap F)$

   $\sigma$$⟶$${\sigma |}_E$

   is an isomorphism, where ${\sigma |}_E$ denotes the restriction of $\sigma$ to $E$.

Remark 1.

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