Degree of a nonconstant morphism of curves

Assume the Axiom of Choice for the cited finiteness route (The Axiom of Choice). Let $k$ be a field and let $f:C\to D$ be a nonconstant morphism of smooth proper geometrically integral curves over $k$ (Curves over a field). By Nonconstant morphisms of proper curves are finite and surjective the morphism $f$ is surjective and finite, so $f$ is dominant, the comorphism $k(D)\to k(C)$, $g\mapsto g\circ f$, is an injective homomorphism of $k$-algebras, and the function-field extension $k(C)\mathrm{/}k(D)$ is finite, (Finitely generated field extensions $F(a_1,\dots ,a_r)$, Function field of an integral finite-type scheme). The degree of $f$ is $\mathrm{deg}(f):=[k(C):k(D)],$ the degree of the finite extension of function fields (The degree $[K:F]={\dim }_{F}K$ of a finite field extension). It is a positive integer.

The degree also has a precise fibre formula. For a closed point $q\in D$, take an affine neighbourhood $V=\mathrm{Spec}R$ and put $A={\mathcal{O}}_{D,q}$ and $B=\mathrm{\Gamma }(f^{-1}(V),{\mathcal{O}}_C){\otimes }_{R}A$. The ring $A$ is a discrete valuation ring with residue field $\kappa (q)$, and $B$ is finite over $A$ because $f$ is finite (Finite morphisms of schemes, A local ring is a nonzero commutative ring with a unique maximal ideal, Local rings at closed points of smooth curves are discrete valuation rings). Since $f$ is dominant and $C$ is integral, $B$ is torsion-free over $A$; hence it is free over the discrete valuation ring $A$ (Every DVR is a PID, Every finitely generated torsion-free module over a PID is free): dominance makes $A\to B$ injective, and $B$ is a domain because $f^{-1}(V)$ is an open subscheme of the integral curve $C$. Its rank is the dimension of its generic fibre $B{\otimes }_{A}k(D)=k(C)$ over $k(D)$, namely $\mathrm{deg}(f)$ (Function field of an integral finite-type scheme, The degree $[K:F]={\dim }_{F}K$ of a finite field extension). Therefore ${\dim }_{\kappa (q)}(B\mathrm{/}{\mathfrak{m}}_{q}B)=\mathrm{deg}(f)$.

The finite-dimensional fibre algebra $B\mathrm{/}{\mathfrak{m}}_{q}B$ is Artinian, and its local factors are indexed by the points $p\in f^{-1}(q)$; the factor at $p$ is ${\mathcal{O}}_{C,p}\mathrm{/}(f^{\ast }{t}_q)$ for a uniformizer $t_q$ of $A$ (Scheme-theoretic fibre, An Artinian ring is canonically the finite product of its localizations at its maximal ideals). The local ring ${\mathcal{O}}_{C,p}$ is a discrete valuation ring. Define $e_p={\mathrm{ord}}_p(f^{\ast }{t}_q)$, the ramification index at $p$; then the local quotient has composition length $e_p$ as an ${\mathcal{O}}_{C,p}$-module: its filtration by powers of a uniformizer has $e_p$ successive quotients, each isomorphic to $\kappa (p)$ (Local rings at closed points of smooth curves are discrete valuation rings, Every nonzero fraction is a unit times a power of a uniformiser, Composition series and length of a module). Since $f$ is finite, $\kappa (p)\mathrm{/}\kappa (q)$ is a finite extension; each composition factor therefore has $\kappa (q)$-dimension $[\kappa (p):\kappa (q)]$. Thus the local factor has $\kappa (q)$-dimension $e_p[\kappa (p):\kappa (q)]$. Additivity of dimension across the local factors gives the weighted fibre formula ${\sum }_{p\in f^{-1}(q)}{e}_p[\kappa (p):\kappa (q)]=\mathrm{deg}(f)\mathrm{.}$

By A finite extension has degree one if and only if the two fields are equal, $\mathrm{deg}(f)=1$ exactly when the function-field inclusion is an isomorphism, which is the definition of birationality (Birational morphisms of integral finite-type schemes). Since $C$ and $D$ are smooth, proper and geometrically integral, a birational morphism between them is an isomorphism (Birational smooth proper curves are isomorphic); conversely an isomorphism induces an isomorphism of function fields and has degree one. Thus $\mathrm{deg}(f)=1⟺f\text{ is birational}⟺f\text{ is an isomorphism},$ and $\mathrm{deg}(f)\ge 2$ whenever $f$ is not an isomorphism. The degree is multiplicative in composites: for nonconstant morphisms $C\to D\to E$ of such curves, the function fields form the finite tower $k(E)\subseteq k(D)\subseteq k(C)$, so multiplicativity follows from the tower law for finite field extensions (Tower law for finite extensions: $[L:F]=[L:K][K:F]$).