We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$. Verify that the family $\left(u^{k}(x)\right)_{0 \leqslant k \leqslant p-1}$ is free. Deduce that $p = 2$.
We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Verify that the family $\left(u^{k}(x)\right)_{0 \leqslant k \leqslant p-1}$ is free. Deduce that $p = 2$.