My attempt:
The normal closure of $E/F$ implies $E/F$ is a normal extension of $F$ over $\mathbb{Q}$. Since the characteristic of $\mathbb{Q}$ is 0, then $\mathbb{Q}$ is a perfect field and every polynomial over $\mathbb{Q}$ is seperable, and so $E/\mathbb{Q}$ is a Galois extension.
I see that the proceeding is fine; but then I have no idea where to go after this. I tried to show that since normal extensions correspond to normal subgroups of the Galois group, then we can assume the degree of $[E:F]=2$ then the index of the corresponding normal subgroup $H$ is $[G:H]=2$. But at this stage I feel like I am headed in the complete wrong direction. Hints?