Let $\phi:C_1\to C_2$ be a nonconstant map of two smooth curves over some algebraically closed field $K$ and let $P\in C_1$. $\phi$ gives us an induced map of fields $\phi^*:K(C_2)\to K(C_1)$, $\phi^*:f \mapsto f\circ \phi$. Silverman defines in his book on elliptic curves that the ramification degree of this map is defined as
$e_\phi(P)=\textrm{ord}_P(\phi^*t_{\phi(P)}),$
where the order is just the normalized valuation of the DVR $K[C_1]_P$ and $t_{\phi(P)}$ denotes a uniformizer of $K[C_2]_Q$, where $Q=\phi(P)$. I'm trying to explicitly write down the connection between this and the corresponding ramification degree in the Dedekind domains, since Silverman seems to hint that they agree.
Let $\mathfrak{m}_Q$ denote the maximal ideal of $A=\phi^*K[C_2]_Q$ and let $B$ be the integral closure of $A$ in $K(C_1)$. Since, $A$ is a DVR, we know that $B$ is Dedekind and we have a factorization
$\mathfrak{m}_QB = \mathfrak{P}_1^{e_1}\cdots \mathfrak{P}_n^{e_n}.$
To get a nice description of the ramification degree in the setting of these Dedekind rings, I would need to show that:
There's a bijection between the primes $\mathfrak{P}_i$ and the points $P\in \phi^{-1}(Q)$.
If $P$ corresponds to some $\mathfrak{P}_i$, then $e_\phi(P)=e_i$.
Does anyone know how this is done? My guess is that both should be simple applications of the Nullstellensatz. I'm getting stuck here with the issue I had in a question I asked yesterday:
Trying to parse a definition in Silverman's EC book
The problem being that I don't know how to concretely compute the map $\phi^*$ and I also have trouble figuring out what $\mathfrak{m}_Q$ should look like. By the Nullstellensatz, we know that it's of the form:
$(X_1-a_1,\ldots,X_n-a_n),$
I take it that in projective space $Q=[a_1,\ldots,1,\ldots,a_n]$, where the $1$ is in position $i$ corresponding to the embedding of $\mathbb{A}^n$ into $\mathbb{P}^n$? But then one would have to express $\phi^*\mathfrak{m}_Q$ explicitly and here one is again left with the problem of concretely writing down what $\phi^*$ is in order to use the Nullstellensatz in $B$.