Ramification index of a morphism of curves ($e_P={\mathrm{ord}}_P({\phi }^{\ast }t)$)

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.