Definition
Let \(f:X\to Y\) be a morphism of smooth curves. Let \(P\in X\) be a point mapping to \(Q=f(P)\in Y\) and let \(t\) be a uniformizer for the DVR \(\mathcal O_{Y,Q}\). Then the ramification index of \(P\) is \[e_P := \nu_P(f^\sharp(t))\] where \(\nu_P\) is the valuation on \(\mathcal O_{X,P}\) and \(f^\#: \mathcal O_{Y,Q}\to \mathcal O_{X,P}\) is the local map of DVRs.