We recently learned in class that if we have a tower of fields
$F \subseteq E \subseteq L$
then $[L:F]$ is finite iff both $[E:F]$ and $[L:E]$ are. In this case we have $[L:F]= [L:E][E :F]$. So then I thought that perhaps it is natural to ask the following question:
If we have a field $F$ and two fields $E$ and $L$ that contain $F$ such that $[E:F] = m$ and $[L : F] = n$, with $n > m$, is it the case that $L$ must contain $E$? Obviously if $L$ contains $E$ then $n > m$ because then $E$ would be an $F$ - vector subspace of $L$.
Thanks.
Edit: I have observed that if we have a field extension that is finite over the reals (Say $L$) then $L$ is algebraic over $\mathbb{R}$ so that it is over $\mathbb{C}$. But then this is only possible when $L = \mathbb{C}$ because $\mathbb{C}$ is algebraically closed.