Formula $\text{deg}f^*D=\text{deg}f\cdot\text{deg}D$ and $n=\sum_i e_i f_i$

Question

In proposition II.6.9 page 138 Hartshorne write $\text{deg}f^*D=\text{deg}f\cdot\text{deg}D$ for $X,Y$ non singular curves (over an algebraically closed field $k$) and $f:X\to Y$ a finite morphism. When I see this formula I think to decomposition in number field with $f:\text{Spec}(A)\to\text{Spec}(\mathbb{Z})$ induced by inclusion hence with $D=\mathfrak{p}=(p)$ the equality $\mathfrak{p}A=\prod_i \mathfrak{q}_i^{e_i}$ becomes $f^*(\mathfrak{p})=\sum_i e_i \mathfrak{q}_i$. I guess there is a generalization of formula II.6.9 that should take into account of the dimensions $[k(\mathfrak{q}_i):k(\mathfrak{p})]=f_i$ and generalize $n=\sum_i e_i f_i$. Do you know this generalization and in what standard refenrence it is studied? Thanks