Is $K/F$ also Galois when $K/k$ is Galois and $k \subset F \subset K$?

Question

If $k\subset F\subset K$ and $K/k$ is Galois (infinite extension) is then $K/F$ also Galois ?

In the case where $K/k$ is finite extension this follows from fundamental theorem but here I don't know.

Answer 1

Yes, this is true. This is proved in Lang, Algebra, combining theorems 3.4 and 4.5.

— On the one hand, $K/F$ is separable. Let $x \in K$ and let $f$ be the minimal polynomial of $x$ over $F$ (we only define separability for algebraic extensions).

Let $g$ be the minimal polynomial of $x$ over $k$. You know that $f$ divides $g$. What do you know about $g$ if $K/k$ is assumed to be separable? What can you conclude about $f$, then?

— On the other hand, $K/F$ is a normal extension. Let $\sigma : K \to \overline k$ be a field morphism that fixes $F$. Then it fixes $k$. Since $K/k$ is normal, what do you know about $\sigma$? What can you conclude about $K/F$ then?