In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $K$.

I am having trouble understanding why there is no inertia degree appearing in the degree formula $\deg(\Phi)=\sum_{P\in \Phi^{-1}(Q)}e_{\Phi}(P)$ for any $Q\in C_{2}$. This is clear in characteristic 0 as $K$ becomes algebraically closed. But I don't see why in general the inertia degree should be 1 and not appear in the well known degree formula as it does in standard Number Theory.

This is what I mean: as the curves are smooth the regular functions rings are Dedekind domains and we have the unique factorization $M_{\Phi(P)}=\prod_{P′\in \Phi^{-1}(\Phi(P))}M_{P'}^{e_{\Phi}(P′)}$ being $M_{\Phi(P)}$ and $M_{P}$ the maximal ideals associated to $\Phi(P)$ and $P$ respectively (as $\Phi$ is non constant $\Phi^{*}$ is injective and embeds $K[C_{2}]$ in $K[C_{1}]$). In the degree formula the inertia degree should appear by standard Number Theory, which is not the case.

In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $K$.

I am having trouble understanding why there is no inertia degree appearing in the degree formula $\deg(\Phi)=\sum_{P\in \Phi^{-1}(Q)}e_{\Phi}(P)$ for any $Q\in C_{2}$. I don't see why in general the inertia degree should be 1 and not appear in the well known degree formula as it does in standard Number Theory.

This is what I mean: as the curves are smooth the regular functions rings are Dedekind domains and we have the unique factorization $M_{\Phi(P)}=\prod_{P′\in \Phi^{-1}(\Phi(P))}M_{P'}^{e_{\Phi}(P′)}$ being $M_{\Phi(P)}$ and $M_{P}$ the maximal ideals associated to $\Phi(P)$ and $P$ respectively (as $\Phi$ is non constant $\Phi^{*}$ is injective and embeds $K[C_{2}]$ in $K[C_{1}]$). In the degree formula the inertia degree should appear by standard Number Theory, which is not the case.

In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $K$. I am having trouble understanding why there is no inertia degree appearing in the degree formula $\deg(\Phi)=\sum_{P\in \Phi^{-1}(Q)}e_{\Phi}(P)$ for any $Q\in C_{2}$. This is clear in characteristic 0 as $K$ becomes algebraically closed. But I don't see why in general the inertia degree should be 1 and not appear in the well known degree formula as it does in standard Number Theory. This is what I mean: as the curves are smooth the regular functions rings are Dedekind domains and we have the unique factorization $M_{\Phi(P)}=\prod_{P′\in \Phi^{-1}(\Phi(P))}M_{P'}^{e_{\Phi}(P′)}$ being $M_{\Phi(P)}$ and $M_{P}$ the maximal ideals associated to $\Phi(P)$ and $P$ respectively (as $\Phi$ is non constant $\Phi^{*}$ is injective and embeds $K[C_{2}]$ in $K[C_{1}]$). In the degree formula the inertia degree should appear by standard Number Theory, which is not the case.

In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $K$. 

I am having trouble understanding why there is no inertia degree appearing in the degree formula $\deg(\Phi)=\sum_{P\in \Phi^{-1}(Q)}e_{\Phi}(P)$ for any $Q\in C_{2}$. This is clear in characteristic 0 as $K$ becomes algebraically closed. But I don't see why in general the inertia degree should be 1 and not appear in the well known degree formula as it does in standard Number Theory. 

This is what I mean: as the curves are smooth the regular functions rings are Dedekind domains and we have the unique factorization $M_{\Phi(P)}=\prod_{P′\in \Phi^{-1}(\Phi(P))}M_{P'}^{e_{\Phi}(P′)}$ being $M_{\Phi(P)}$ and $M_{P}$ the maximal ideals associated to $\Phi(P)$ and $P$ respectively (as $\Phi$ is non constant $\Phi^{*}$ is injective and embeds $K[C_{2}]$ in $K[C_{1}]$). In the degree formula the inertia degree should appear by standard Number Theory, which is not the case.

In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $K$. I am having trouble understanding why there is no inertia degree appearing in the degree formula $\deg(\Phi)=\sum_{P\in \Phi^{-1}(Q)}e_{\Phi}(P)$ for any $Q\in C_{2}$. This is clear in characteristic 0 as $K$ becomes algebraically closed. But I don't see why in general the inertia degree should be 1 and not appear in the well known degree formula as it does in standard Number Theory. This is what I mean: as the curves are smooth the regular functions rings are Dedekind domains and we have the unique factorization $M_{\Phi(P)}=\prod_{P′\in \Phi^{-1}(\Phi(P))}M_{P'}^{e_{\Phi}(P′)}$ being $M_{\Phi(P)}$ and $M_{P}$ the maximal ideals associated to $\Phi(P)$ and $P$ respectively. In the degree formula the inertia degree should appear by standard Number Theory, which is not the case.

In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $K$. I am having trouble understanding why there is no inertia degree appearing in the degree formula $\deg(\Phi)=\sum_{P\in \Phi^{-1}(Q)}e_{\Phi}(P)$ for any $Q\in C_{2}$. This is clear in characteristic 0 as $K$ becomes algebraically closed. But I don't see why in general the inertia degree should be 1 and not appear in the well known degree formula as it does in standard Number Theory. This is what I mean: as the curves are smooth the regular functions rings are Dedekind domains and we have the unique factorization $M_{\Phi(P)}=\prod_{P′\in \Phi^{-1}(\Phi(P))}M_{P'}^{e_{\Phi}(P′)}$ being $M_{\Phi(P)}$ and $M_{P}$ the maximal ideals associated to $\Phi(P)$ and $P$ respectively (as $\Phi$ is non constant $\Phi^{*}$ is injective and embeds $K[C_{2}]$ in $K[C_{1}]$). In the degree formula the inertia degree should appear by standard Number Theory, which is not the case.