corollary to the compositum of a Galois extension and another extension is Galois

Corollary 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 $\mathrm{[}EF\mathrm{:}F\mathrm{]}\mathrm{=}\mathrm{[}E\mathrm{:}E\mathrm{\cap }F\mathrm{]}$.

This follows immediately from item (2) of the theorem.