Let $F\in\mathbb{F}_q[X]$. I.e. a polynomial with coefficients in $\mathbb{F}_q$. Then there is an obvious way to define the degree: the highest power of $x$ with a non-zero coefficient. There are however, two different ways to define it. The first as $$\deg(F):= \dim_{\mathbb{F}_q}(\mathbb{F}_q[x]/(F)).$$ The second one is if $P_\infty$ is the prime at infinity of $\mathbb{F}_q(X)$, then $$\deg(F)= -v_{P_\infty}(F).$$
If we replace $\mathbb{F}_q(X)$ with a finite extension of $\mathbb{F}_q(X)$, $L$, then if we have $F\in\mathcal{O}_L$, the degree of $F$ is $$\deg(F):=\dim_{\mathbb{F}_q}(\mathcal{O}_L/F\mathcal{O}_L)$$
Now, my question is, how would we write this in terms of valuations of prime at infinity. If $\mathfrak{P}_1,\dots,\mathfrak{P}_n$ are the primes dividing $P_\infty$, then would it be that $$\deg(F) = -\sum_{i=1}^n v_{\mathfrak{P}_i}(F)?$$
Or, would we have to incorporate ramification information? That is, if $$P_\infty \mathcal{O}_L = \prod_{i=1}^n \mathfrak{P}_i^{e_i}$$ then would $$\deg(F) = - \sum_{i=1}^n e_iv_{\mathfrak{P}_i}(F)?$$
Any solution or reference would be greatly appreciated.