I've been working on some exercises on Galois theory and I'm a little stuck in one of them: Let $L$ be the splitting field of the polynomial $x^3-10$ (so $L=\mathbb{Q}(\sqrt[3]{10},\zeta)$ where $\zeta$ is complex cubic root of unity) and $K=\mathbb{Q}(\sqrt{5},\sqrt{-7})$. I need to compute the degree of $K\cap L$, $|K\cap L:\mathbb{Q}|$. I computed the degrees of $K$ and $L$ and they are $4$ and $6$ respectively, so $|K\cap L:\mathbb{Q}|$ must be $1$ or $2$. It probably is $1$ but I can't find any good reason, so I need some hints.
Thanks in advance.
Hint:
If $[K \cap L : \mathbb{Q}]=2$, then $K$ and $L$ share a quadratic extension of $\mathbb{Q}$. Now $L$ only has one quadratic subfield whereas $K$ has three - can you work out what these subfields are explicitly (ie write them uniquely as $\mathbb{Q}(\sqrt{d})$ with $d \in \mathbb{Z}$ squarefree)?