= Degree of a morphism of curves
{title2=$\deg f=[k(C):k(D)]$}
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 $\deg f$. 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.
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