Let $F/K$ be an algebraic function field, with constant field $K=K_{F}$, and $L/F$ be an finite extension.
Is it possible that infinitely many places of $L$ ramifiy?
Somehow I feel that it is not possible because $L/F$ is finite, but I am not very clear why. I would appreciate if anyone can provide some suggestions.
In an inseparable extension, every place is ramified. If the extension is separable, then only finitely many places ramify.
One can argue as follows: Think of this geometrically, i.e. we have a finite morphism of curves $D\to C$, then one can show that the support of $\Omega^1_{D/C}$ consists exactly of the ramified primes, but this a closed subset and a proper subset if the function field extension is separable, thus it consists of finitely many points.
This whole argument also works in the more general setting of Dedekind schemes, e.g. $\mathrm{Spec}(\mathcal O_K)$ for a number field $K$.