Existence of finite Galois extension

Question

Let $L/F$ be an infinite Galois extension. Then to any finite extension $F\subset K\subset L$, why is there always a finite extension of $K$, say $E$, s.t. $E/F$ is Galois?

Answer 1

Since $L/F$ is separable all subfields all automatically separable over $F$ as well.

In addition, for an extension $K/F$ we may construct the normal closure $\overline{K}$ of $K$ in $L$ (since $L$ is normal), which is a normal and separable extension of $F$, hence Galois.

Finally, $\overline{K}/F$ is a finite extension because $K$ is finite over $F$ so that $K = F(\alpha)$ for some primitive element $\alpha$ and we just need to adjoin all Galois conjugates of $\alpha$ into $\overline{K}$.