I'm reading The Arithmetic of Elliptic Curves, by J. Silvermann, 2nd edition. I don't understand the following proof.
Why is this homomorphism between $E_{1}[m] \to E_{2}[m]$ a isomorphism?
I know that if $\phi$ is separable, then $\#ker\phi = deg(\phi)$. Then, I have that the kernel no contains points of order m (Remembering that (m,deg($\phi))=1$. So, we are done.
But, $\phi$ can have inseparable part. So I don't understand how I can conclude that it will be always an isomorphism.