Degree of a morphism of curves ($\mathrm{deg}f=[k(C):k(D)]$)

A nonconstant morphism of algebraic varieties $f:C\to D$ between smooth projective integral curves is finite, and its degree is the degree of the induced extension of function fields. It is also the degree of the pullback of a rational point, with ramification multiplicities and residue degrees included: locally the finite coordinate algebra is a torsion-free module over the target discrete valuation ring, hence free of its generic rank. For $D={\mathbb{P}}^1$ and a rational function defining $f$, the pullback of infinity is its pole divisor, so the total pole degree equals $\mathrm{deg}f$. A degree-one finite map to a normal curve is an isomorphism, since a finite integral algebra inside its unchanged function field equals the integrally closed target ring.