Lemma 2

Let D be a derivation of E such that $D^p = D$ and let K be the kernel of D. Suppose that there exists a non-zero x in E such that $D(x) = x$. Then K is a subfield of E containing $E^p$ and we have $[E : K] = p$.

It is clear that K is a subfield of E containing $E^p$. Denote by $K_i$ the kernel of $D - i$ for $i \in \mathbf{F}_p$. We have $K_0 = K$ and Lemma 1 shows that $E = \bigoplus_{i \in \mathbf{F}_p} K_i$. Let $i \in \mathbf{F}_p$ and let $u$ be in $K_i$; we have

whence $xu \in K_{i+1}$. Since x is non-zero, multiplication by x is an automorphism of the vector K-space E which maps $K_i$ onto $K_{i+1}$ for all $i \in \mathbf{F}_p$. Since $[K_0 : K] = 1$, we have $[K_i : K] = 1$ for all $i \in \mathbf{F}_p$ whence $[E : K] = p$.