= Ramification index of a morphism of curves
{title2=$e_P=\operatorname{ord}_P(\varphi^*t)$}
= Ramification indices of a morphism of curves
{synonym}
For a nonconstant morphism between nonsingular curves and a local target <uniformizer> $t$ at the image of $P$, the index is the <order of vanishing> of its pullback. Changing $t$ multiplies it by a local <unit in a ring> and does not change that order. This is a <valuation> definition and remains meaningful for inseparable morphisms.
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