Suppose $\Bbb{Q} \subseteq F \subseteq \Bbb{C}$ is a tower of extensions and suppose that $i \in F$.
If the extension $\Bbb{Q} \subseteq F$ is finite, show that $[F : \Bbb{Q}]$ is even.
What I have attempted so far:
By the tower law we have
$$ [\Bbb{C} : \Bbb{Q}]= [\Bbb{C} : F][F : \Bbb{Q}]$$
Now $ [\Bbb{C} : \Bbb{Q}]=\infty$ and so since $[F : \Bbb{Q}]$ is finite we must have $[\Bbb{C} : F]=\infty$ now I don't know how to proceed any advice?