Theorem 2.8. Assuming that:

Then

\sum_{P \in ϕ^{- 1} Q} e_{ϕ} (P) = \deg ϕ \forall Q \in C_{2} .

Moreover, if $ϕ$ is separable then $e_{ϕ} (P) = 1$ for all but finitely many $P \in C_{1}$ . In particular:

(i) $ϕ$ is surjective (on $\bar{K}$ -points)
(ii) $# ϕ^{- 1} (Q) \leq \deg ϕ$
(iii) If $ϕ$ is separable then equality holds in (ii) for all but finitely many $Q \in C_{2}$ .