I have the following problem:
Let $p(x)=(x^{12}-16)(x^2-3)$. Show that $K=\mathbb{Q}(\sqrt[3]{2},\sqrt{3},i)$ is the splitting field of $p$ over $\mathbb{Q}$, $[K:\mathbb{Q}]=12$ and show that exists a Galois extension $E$ of $\mathbb{Q}$ such that $\mathbb{Q}<E<K$ and $[E:\mathbb{Q}]=6.$
I've done the first two parts already. For the third, I've tried to get explicitly the field $E$ but i haven't succeded. If you draw the diagram you have two options for $E$, $\mathbb{Q}(\sqrt[3]{2},i)$ or $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$. The problem is that I can't conclude with these two options.
What I'm missing?
Thanks in advance and sorry for my bad english.