Corollary 6.4:Arithmetic of Elliptic Curves,Silverman

Question

Notation:$\hat{\phi}$ is the dual isogeny for the Frobenius morphism($\phi$).

In proving (c) part of this corollary,we have 2 cases.Either $\hat{\phi}$ is separable or inseparable.Suppose $\hat{\phi}$ is inseparable,then it is stated that deg$_{s}\hat{\phi}$=1.But this is true if $\hat{\phi}$ is purely inseparable.Is purely inseparable and inseparable the same thing?

Answer 1

$\deg_s(\hat{\phi})$ divides $\deg(\hat{\phi}) = p$, so $\deg_s(\hat{\phi})$ either 1 or , i.e., $\hat{\phi}$ is either purely inseparable or separable.