Let $g(x)$ of degree 4 be an irreducible polynomial in $\Bbb Q$. Let $\alpha \in \Bbb C$ be a root of $g(x)$, and let $R=\Bbb Q(\alpha)$. We also let $K$ be the splitting field of $g(x)$ over $R$. Further, assume that there exists a quadratic field $L$ such that $\Bbb Q \subset L \subset R $, and that $[L:\Bbb Q]=2$, and assume that $K$ properly contains $R$ (i.e. $R \neq K$).
Prove that $[K:\Bbb Q]=8$ and find $Gal(K/\Bbb Q)$.
I have found that $Gal(K/\Bbb Q)$ needs to be a transitive subgroup of $S_4$, so then the options are $S_4, A_4, D_4$, Klein-Four subgroup, $\Bbb Z/4\Bbb Z$.
At this point, I would want to show that $[K:L]=4$ implying the first result. The second result most likely comes from using the resolvent cubic, but I don't see how to use it.
Any hints or help is appreciated, thanks