# Let $\phi$ be the Frobenius morphism; why is $1-\phi$ separable? [closed]

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?

• 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. – R. van Dobben de Bruyn May 22 at 18:53
• @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? – danihelovick May 22 at 19:42
• There is a two hours older version of this post at MSE. – KReiser May 22 at 23:50
• @R.vanDobbendeBruyn's point compares to, not cites, the situation with Artin–Schreier extensions; the entire proof is in the comment. – LSpice May 23 at 2:46