Degree divisor proper curve

Let $k$ be a field. A proper curve over $k$ is an integral $k$-scheme $C$ (Integral schemes) whose structure morphism $C\to \mathrm{Spec}k$ is proper (Proper morphisms) and whose underlying topological space has chain dimension one (Chain dimension and the empty-space convention). Thus $C$ is of finite type over $k$. No normality, regularity, projectivity, or smoothness is assumed.

For a closed point $x\in C$, its residue field $\kappa (x)$ (The residue field at a point of an affine scheme) is finite over $k$. Indeed, choose an affine open neighborhood $U=\mathrm{Spec}A$ of $x$ (Affine schemes and their coordinate rings). Since $C\to \mathrm{Spec}k$ is of finite type, $A$ is a finite-type $k$-algebra; the closed point $x$ corresponds to a maximal ideal $\mathfrak{m}\subset A$, so $\kappa (x)\cong A\mathrm{/}\mathfrak{m}$ is finite over $k$ (A maximal ideal of an affine algebra has finite residue field over the base field). Write $[\kappa (x):k]={\dim }_k\kappa (x)$.

We use divisor on $C$ to mean a finite formal integral linear combination of closed points, $D={\sum }_{x\in C\text{ closed}}{n}_x[x],n_x\in \mathbb{Z},$ where only finitely many $n_x$ are nonzero. These divisors form the free abelian group $\mathrm{Div}(C)$ on the closed points. Define the $k$-degree by ${\mathrm{deg}}_{k}D={\sum }_{x\in C\text{ closed}}{n}_x[\kappa (x):k]\in \mathbb{Z}\mathrm{.}$ This is well-defined because the support is finite, and coefficientwise addition makes ${\mathrm{deg}}_k:\mathrm{Div}(C)\to \mathbb{Z}$ a group homomorphism.