Residue field on $\mathrm{spec} ( \mathbb{Z})$ Let $K$ be a field, and $X$ a regular projective curve over $\mathrm{Spec} ( K)$. For any closed point $x$ we have a residue field, denoted $\kappa(x)$, which is a finite extension of my ground field $K$. And we denote $[\kappa(x):K]$ the residue degree on $x$.
Now let consider $\mathrm{Spec}\mathbb(Z)$, the closed points are the prime numbers $p$, which have for residue field the field $\mathbb{F}_p$. Do we have an analogue of the residue degree on the prime $p$ ?
Thanks for help !
 A: Recall that the residue degree of a closed point $x \in \mathrm{Spec}(\mathbb F_q[T])$ is $\log_q |\kappa(x)|$.  In $\mathrm{Spec}(\mathbb Z)$ we have no fixed prime power $q$, so it's natural (no pun intended) to define the residue degree of $x \in \mathrm{Spec}(\mathbb Z)$ as the natural logarithm of $|\kappa(x)|$; that is, the point $[p]$ has degree $\log p$.
This is actually a standard idea in Arakelov theory.  Namely, define an Arakelov divisor on $\mathrm{Spec}(\mathbb Z)$ to be a pair $(D, g)$, where $D = (n_p)_{p \text{ prime}}$ is a divisor on $\mathrm{Spec}(\mathbb Z)$, and $g$ is a real number which we view as encoding data at the archimedean place of $\mathbb Z$.  Just as a nonzero rational function on a curve induces a divisor, a nonzero rational number $a$ induces an Arakelov divisor
$$\widehat{\mathrm{div}}(a) := (\mathrm{div}(a), -\log |a|).$$
Then, just as one can take the degree of a divisor on a curve, there is a notion of the arithmetic degree of an Arakelov divisor:
$$\widehat{\mathrm{deg}}(D, g) := g + \sum_p n_p \cdot \log p.$$
Here the $\log p$ factor plays the role of the usual degree of a point (of a curve over a field).  You can check that every principal Arakelov divisor (i.e. each $\widehat{\mathrm{div}}(a)$ for $a \in \mathbb Q^{\times}$) has arithmetic degree zero.  This is analogous to the fact that a rational function on an algebraic curve has as many zeroes as poles (counted with multiplicity).  It's an early instance of the philosophy that the nonarchimedean place gives a "completion" of the affine curve $\mathrm{Spec}(\mathbb Z)$.
Reference for the Arakelov theory:  section 1.2.B of Chambert-Loir.
