Hello, this was another problem that I had difficulty with:

Let $\displaystyle F$ be a field extension of $\displaystyle K$, and let $\displaystyle L$ be a closed intermediate field. Let $\displaystyle G=Aut_kF$. Show that the normalizer of $\displaystyle L'$ in $\displaystyle G$ is the set of elements of $\displaystyle G$ which map $\displaystyle L$ onto itself.

$\displaystyle L'$ is the corresponding subgroup (in $\displaystyle G$) of $\displaystyle L$.

I'm not sure. I wanted to use FTGT (Fundamental Theoreom of Galois Theory), but this problem doesn't state that the extension is finite.