Let $C$ be an elliptic curve defined over $\mathbb{F}_p$. We know that the Frobenius morphism $\phi:C\longrightarrow C$ is inseparable, i.e, $K(C)/\phi^*K(C) = K(C)/K(C)^q$ is an inseparable extension.

How can I show that $1-\phi$ is separable?

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    $\begingroup$ For the same reason that Artin–Schreier extensions are separable: on tangent spaces, $\phi$ acts as the $0$ map, so $1-\phi$ acts as the identity. $\endgroup$ – R. van Dobben de Bruyn May 22 at 18:53
  • $\begingroup$ @R.vanDobbendeBruyn thank you, but i have not knowledge about Artin–Schreier extensions and not about the action of a morphism on tangent spaces. So, could you recommend me a reference about it? $\endgroup$ – danihelovick May 22 at 19:42
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    $\begingroup$ There is a two hours older version of this post at MSE. $\endgroup$ – KReiser May 22 at 23:50
  • $\begingroup$ @R.vanDobbendeBruyn's point compares to, not cites, the situation with Artin–Schreier extensions; the entire proof is in the comment. $\endgroup$ – LSpice May 23 at 2:46