I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number fields?
Given a compact Riemann surface $X$ and a finite cover $$\pi : X \to \mathbb{P}^1$$ we get an induced map on the function fields, $$\pi ^* : \mathbb{C}(\mathbb{P}^1) \to \mathbb{C}(X)$$ this turns out to be an inclusion, so we get a finite field extension $$\mathbb{C}(X) / \mathbb{C}(\mathbb{P}^1)$$
We say a point $P \in X$ is ramified if locally, $\pi$ looks like $z\mapsto z^e$ for some $e>1$.
We say a prime ideal $(p)\subset\mathbb{C}(\mathbb{P}^1)$ ramifies if it has a factorisation in $\mathbb{C}(X)$, $(p)=\mathfrak{p}_1^{e_1}...\mathfrak{p}_r^{e_r}$, where any one of the $e_i$ is $>1$.
But $(p)$ corresponds to an element of $\mathbb{P}^1$, so to be consistent, would it not make more sense to say $\mathfrak{p}_1^{e_1}$ ramifies, and that $(p)$ is a branch point?
Thanks in advance,