Equality of divisors under Galois action in Silverman

Question

I am trying to understand a formula in Silverman's The Arithmetic of Elliptic Curves. Let $K$ be a field, $\overline{K}$ an algebraic closure of $K$, and $C$ be a curve defined over $K$, and $\sigma \in G_{\overline{K} / K}$. Then for $f \in \overline{K}(C)^*$, we have $$\mathrm{div}(f^\sigma) = (\mathrm{div}(f))^\sigma.$$ I think the reasoning is that $$(\mathrm{div}(f))^\sigma = \sum_{P \in C} \mathrm{ord}_P(f) P^\sigma,$$ while \begin{align*} \mathrm{div}(f^\sigma) & = \sum_{P \in C} \mathrm{ord}_P(f^\sigma) P \\ & = \sum_{P \in C} \mathrm{ord}_{P^\sigma} (f^\sigma) P^\sigma && \text{by reindexing} \\ & = \sum_{P \in C} \mathrm{ord}_{P}(f) P^\sigma && \text{since $f^\sigma(P^\sigma) = (f(P))^\sigma$.} \end{align*} I'm not sure that the reindexing step is valid, however, because I don't think that $\sigma$ necessarily acts bijectively on the points of $C$. What am I missing here?

Answer 1

The reindexing works because you can partition $C$ into its Galois orbits and $G_{\bar{F}/F}$ acts transitively on each orbit.