Let $E/F$ be a Galois extension, and let $B$ be an intermediate field between $E$ and $F$. Let $H$ be the subgroup of $Gal(E/F)$ that maps $B$ into itself (but does not necessarily fix $B$). Prove that $H$ is the normalizer of $Gal(E/B)$ in $Gal(E/F)$.
I approached this by showing inclusion in two directions. I have managed to show that $H$ is a subgroup of the normalizer of $Gal(E/B)$ but am unsure how to show the other direction. I think I would need to show that an element of the normalizer maps $F$ into $F$, but don't know how to do this. Any suggestion is greatly appreciated! (So I would like to show that the image of F under an element of the normalizer is a subset of F)