I am having much difficulty proving the following fact.

Let $\phi:C_1\to C_2$ be a Galois cover of curves over an algebraically closed field $K$. Let $G=\mathrm{Gal}(K(C_1)/\phi^*K(C_2))$. Let $N: K(C_1)\to \phi^*K(C_2)$ be the field norm. Let $\phi_*: \mathrm{Div}(C_1)\to \mathrm{Div}(C_2)$ and $\phi^*: \mathrm{Div}(C_2)\to \mathrm{Div}(C_1)$ be the pushforward and pullback maps induced by $f$ respectively. For $f\in K(C_1)$, I wish to show that

$\phi_*(\mathrm{div}(f)) = \mathrm{div}(N(f))$.

I was given the hint that since for $P\in C_1$, $\phi^*\circ \phi_*([P]) = \sum_{\sigma\in G} [\phi(\sigma(P))]$ and $\phi^*$ is injective on divisors it suffices to prove that $f^*\circ f_*(\mathrm{div}(g)) = f^*(\mathrm{div}(N(g)))$. But I have no idea how to show this.

Any help would be greatly appreciated.


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