Galois relations between subfields

Question

$\newcommand{\Gal}{\mathrm{Gal}}$ Suppose I have a field $L$ which is a Galois p-extension over a smaller field, $K$. Suppose further that $K$ is a p-extension over $F$ (a p-extension is a Galois extension with a Galois group of order $p^a$ for some $a \in \Bbb Z_+$. To prove that $L$ is a p-extension over $F$ I am finding it necessary to assume that $K$ is fixed by all the automorphisms $\sigma_i \in \Gal(\overline L/F)$, where $\overline L$ is the Galois closure of $L$ over $F$, i.e., $\sigma_i(K) = K$ for all $\sigma_i \in \Gal(\overline L/F)$. Why is this necessary?

My thanks for any advice you can give me.

Answer 1

Think about $F$ the rationals, $K$ the extension by $\sqrt2$, $L$ the extension by $\root4\of2$. Then $L$ is not Galois over $F$, so you have to go to the Galois closure.

Answer 2

Why don't you use the tower of fields degree theorem?

$$[L:F]=[L:K][K:F]=p^ap^b=p^{a+b}$$

and we're done...!