I have been thinking on the following for a while and can't seem to crack it.
Let $F, F_1, F_2$ be function fields of one variable over a fixed perfect field, $F_i$ contain $F$, and $P_1, P_2$ are places of $F_1,F_2$ such that $P_1\cap F=P_2\cap F$. Also, assume $[F_1F_2:F_2]=[F_1:F]$. Show that there is a place of $F_1F_2$ above $P_1$ and $P_2$.
If $F_1F_2/F$ isn't separable we can first prove for the separable part, and then there is a unique lift to $F_1F_2$. So, assume everything is separable.