Suppose we have a tower of field extensions:
$\overline{F} \subset K \subset E \subset F$
Is it true in general that $|G(K/F)| = |G(K/E)| \cdot |G(E/F)|$?
I was able to verify some specific examples, like $\mathbb{Q}(\sqrt[3]{2}, \omega)$ for $x^3-2$ and another extension, but how could I show that this holds in general for all such towers of extensions?
This is generally not the case. For instance, take $K = \Bbb Q(\sqrt[4]{2}), E = \Bbb Q(\sqrt{2}), F = \Bbb Q$. Then $K/E$ and $E/F$ are both Galois extensions with $|G(K/E)| = |G(E/F)| = 2$. However, the extensions $K/F$ is not Galois, with there being only two automorphisms of $K$ fixing $F$, since such an automorphism is determined by its action on $\sqrt[4]{2}$, which can only be sent to $\pm \sqrt[4]{2}$, the only roots of $x^4-2$ lying in $K$. Thus, $|Aut(K/F)| = 2$.
If you interpret $|G(K/F)|$ to be the order of the Galois group of the Galois closure of $K$ over $F$, then the order is 8 instead, as the Galois closure, $\Bbb Q(\sqrt[4]{2}, i)$ is a degree 8 extension.