Divisor degree with residue degrees over a nonclosed field

Residue field of $x$. On the chart $U_0=\mathrm{Spec}\mathbb{R}[t]$ the point $x=V(t^2+1)$ corresponds to the maximal ideal $(t^2+1)$, which is maximal because $t^2+1$ is irreducible over $\mathbb{R}$ (it has no real root and degree two); hence $\kappa (x)=\mathbb{R}[t]\mathrm{/}(t^2+1)\cong \mathbb{C}$ [F3], a finite extension of $\mathbb{R}$ of degree $2$.

Order of vanishing at $x$. In the local ring ${\mathcal{O}}_{C,x}=\mathbb{R}[t{]}_{(t^2+1)}$ the element $t^2+1$ generates the maximal ideal, hence is a uniformizer and ${\mathrm{ord}}_x(f)=1$: the divisor of $f$ has the term $+[x]$.

Order of the pole at infinity. In the chart $U_{\mathrm{\infty }}=\mathrm{Spec}\mathbb{R}[u]$ with $u=1\mathrm{/}t$ one has $f=t^2+1=u^{-2}(1+u^2)$ with $1+u^2$ a unit of the local ring at $u=0$ because it evaluates to $1$ there; hence ${\mathrm{ord}}_{\mathrm{\infty }}(f)=-2$ and the divisor of $f$ has the term $-2[\mathrm{\infty }]$.

Degree of $[x]$. By the degree formula of [F2], ${\mathrm{deg}}_{\mathbb{R}}([x])=1\cdot [\kappa (x):\mathbb{R}]=1\cdot 2=2$, while the support of $[x]$ is the single point $x$; this is the sense in which the degree counts with residue-field degrees rather than with a point count.

Principal divisor. At every other closed point of $U_0$, $t^2+1$ is a unit, since its only irreducible factor is $t^2+1$; the complement of $U_0$ is the single point $\mathrm{\infty }$. Thus steps 1.2 and 1.3 account for every nonzero order: $\mathrm{div}(f)=[x]-2[\mathrm{\infty }]$, and its degree is ${\mathrm{deg}}_{\mathbb{R}}([x])-2{\mathrm{deg}}_{\mathbb{R}}([\mathrm{\infty }])=2-2\cdot 1=0$ by [F2]; the point $\mathrm{\infty }$ has residue field $\mathbb{R}$ and degree one. Thus $[x]$ is linearly equivalent to $2[\mathrm{\infty }]$, a divisor of the same degree $2$.

Base change to $\mathbb{C}$. On the base-changed affine chart, the fibre of $x$ has coordinate algebra $\mathbb{C}[t]\mathrm{/}(t^2+1)=\mathbb{C}[t]\mathrm{/}((t-\mathrm{i})(t+\mathrm{i}))$. Evaluation at $\mathrm{i}$ and $-\mathrm{i}$ identifies this algebra with $\mathbb{C}\times \mathbb{C}$: every class has a unique representative $a+bt$, and its evaluations $a+b\mathrm{i},a-b\mathrm{i}$ determine $a,b$ uniquely. Thus the fibre consists of the two distinct reduced points $t=\mathrm{i}$ and $t=-\mathrm{i}$, each with residue field $\mathbb{C}$ and degree one. At each point $t^2+1$ has order one, since its other linear factor is a unit. Hence the base-changed divisor is $[\mathrm{i}]+[-\mathrm{i}]$, of degree $2={\mathrm{deg}}_{\mathbb{R}}([x])$.

Conclusion. On ${\mathbb{P}}_{\mathbb{R}}^1$ the divisor $D=[x]$ has degree $2$ although it is supported at one point, its class is the class of $2[\mathrm{\infty }]$ by the principal divisor $[x]-2[\mathrm{\infty }]$, and after base change to $\mathbb{C}$ it becomes the sum of the two degree-one points $\mathrm{i},-\mathrm{i}$ with the same total degree. Degree is therefore computed with residue-field degrees, as in [F2]; Choice in [F4] is inherited from the projective-line and DVR suppliers, whereas additivity of the degree is choice-free and no further selection is used here.