Let $L/k $ be a Galois extension. If $F$ and $F'$ are intermediate fields of $L/k$ then
$L^{⟨Gal(L/F),Gal(L,F’)⟩}=F \cap F'$
($⟨G;G′⟩$ is used to denote the subgroup generated by $H ∪ H′.$)
Using some theorems we have that
$F=L^{Gal(L/F)}$ ; $F'= L^{Gal(L,F’)}$
$ L^{Gal(L/F)} \cap L^{Gal(L,F’)}$
What do I need to do to connect the dots to finish the question or is it already finished and I didn't notice.