Throughout I would like to work over an algebraically closed field of characteristic 0 (so no separability issues), say $k$. My question is the following:
Do there exist two curves $X$ and $Y$ and a necessarily finite morphism $f:X \rightarrow Y$ such that $[k(X):k(Y)] > \text{max}_{P \in X}\{e_P\}$ where $k(X)$ denotes the function field of $X$ and $e_P$ denotes the ramification index of the map at a point $P \in X$?
My thoughts/attempts so far: Hurwitz's Theorem tells us that for a finite separable morphism of curves $f:X \rightarrow Y$ we have $2g(X)-2=(\text{deg }f)(2g(Y)-2)+\text{deg }R$ where $\text{deg }f=[k(X):k(Y)]$, $g(X)$ is the genus of $X$ and $R$ is the ramification divisor. That is, $\text{deg }R=\sum_{P\in X} (e_P-1)$.
Using this I can note that a simple example may come from a morphism between two genus 0 curves that gives a field extension of degree 3 and ramifies at 4 points. In this case each ramification index would have to be 2 and hence the condition would be satisfied. I thought I had an example of higher genus using the double cover of the projective line over $k$ by an elliptic curve, however at one of the four points of ramification the index would have to be at least equal to the field extension degree.
A related extension to this question is whether it is possible to have such a map that any possible ramification type occurs - that is, given two fixed genus values (necessarily $g(X)\geq g(Y)$ by Hurwitz) and $[k(X):k(Y)]$ fixed, can a morphism be constructed that ramifies at any allowable number of points with indices? Now I do not insist on the ramification indices all being smaller than the degree of the field extension. By allowable here I mean that deg $R$ is controlled by the fixed values by Hurwitz's Theorem, and so the greatest number of ramification points is deg $R$ each with index 2, but other partitions of deg $R$ with integers $\geq 2$ are possible.
Thanks,
Andrew