Galois subextensions in a Galois extension

Question

Let $F \subset E \subset L$ be fields such that $L/E$ and $E/F$ are both Galois extensions. Is $L/F$ necessarily a Galois extension?

Answer 1

Hints :

• Any (well behaving) degree $2$ extension is Normal.

• Not all degree $4$ extensions are Normal.

Answer 2

I got it.

$F = \mathbb{Q}$, $E = \mathbb{Q}(\sqrt2)$ and $L = \mathbb{Q}(2^{1/4})$ constitute a counterexample. Thanks a lot Praphulla and Jyrki for the hints.