Let $E/\mathbb{Q}$ be a finite field extension. Let $F,K \subset E$ subfields which contains $\mathbb{Q}$. Let M the smallest subfield of E which contains $F$ and $K$. If $K/\mathbb{Q}$ is a Galois extension then $M/F$ and $K/(K\cap F)$ are Galois extensions and \begin{equation} r : Gal(M/F) → Gal(K/(K ∩ F))\\ σ → σ|_K \end{equation} is a well defined homomorphism.
Can you help me to prove it?
$K/(K\cap F)$ is a Galois extension since $K\cap F$ is an extension of $\Bbb Q$. Then you essentially copy the proof of the diamond isomorphism theorem from Dummit and Foote to show that the homomorphism $Aut(M/F) → Gal(K/(K ∩ F)), σ \mapsto σ|_K $ is well-defined, and an isomorphism. By considering the degrees of the extensions, you conclude that $M=FK$ is a Galois extension of $F$.
Since you
just want a hint, this should be enough; let me know if you get stuck somewhere.