(Sorry for the weird title, I really don't know how to describe this question in a line)
I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all the subject, and I was trying to get some sketchy introduction to the geometrical perspective. I got the book "Codes and Curves" by Judy Walker and I am trying to make the translation between definitions and objects in each book.
Rosen define divisors on a function field $F$ over a base field $k=\mathbb{F}_q$ to be the free abelian group over the set of all primes, and Walker's definition for a divisor on a projective, non-singular algebraic curve $f(x,y)=0$ is similar but not exactly the same, she defines it as the free abelian group over the set of "points with multiplicity"- if we have a point $P$ on the curve we're looking at its orbit under the froubenius map $x\mapsto x^q$, and this set is the "point with multiplicity".
Also, the principal divisor of an element $a\in F$ is defined (in Rosen's book) by $\sum_P v_P (x)P$ where $P$ are the primes of the function field, and $v_P$ are the valuations of these primes. Walker's definition is of divisors of rational functions $h/g$ of the curve, and it is done by taking the intersection divisors of $C_f \cap C_g, C_f\cap C_h$ and substracting them (where $C_f,C_g,C_h$ are the curves obtained by the zero locus of the equations of $f,g,h$). Finally, the definitions for the degree of a prime / point are different - the degree of $P$ is $[O_P/PO_P:k(x)]$ (where $O_P$ is the place of $P$) and the degree of the point $P$ on the curve is the size of its orbit under the Frobenius map $x\mapsto x^q$.
How each pair of definitions is connected to each other? (Thanks for taking the time to reading this!)
EDIT:
After some time I think I got this. For anyone who might be interested in the future, here are my uncertified thoughts about the big picture here. I'll be greatfull for corrections. For simplicity assume that the projectivization of $f(x,y)$, $F(x,y,z)$, gives a non-singular projective curve. Then we can look at the curve in $\mathbb{P}^2(\overline{k})$ that $F$ defines. There is an action of $Gal(\overline{k}/k)$ on this curve. This group is generated by the Frobenius homomorphism. The orbit of each point $P$ is of size $[k(P):k]$. Now look at the homomorphism $\psi_P:k[x,y]/f \to \overline{k}$ which maps $g(x,y)\mapsto g(P)$. The kernel is a prime ideal, hence maximal ($k[x,y]/f$ is a dedekind domain under the above assumptions). Since the image of $\psi_P$ is $k(P)$, the degree of $M$ is $[k(P):k]$. Thus the degrees here are really the same. To finish the big picture here, I claim that the ideal $M$ depends solely on the orbit. Indeed, if we take any other point in the orbit, $P'$, then $\psi_{P'}$ is the composition of $\psi_{P}$ with an isomorphism of $k(P)$ and $k(P')$ over $k$, which we can extend to a an element of $Gal(\overline{k}/k)$. Thus they have the same kernel.
With regard to divisors, I still do not fully understand it; though by thinking of elements in $F$ as functions on the curve, reduction mod a prime $P$ is equivalent to asking wether the function vanish, and then to ask its order of vanishing at $P$. Same about poles. But I really do not understand it computationally.