Corollary 4

— Suppose the field K is of characteristic zero; then the endomorphism u is nilpotent if and only if $\operatorname{Tr}(u^s) = 0$ for $1 \leq s \leq n$.

If $u$ is nilpotent then the $a_i$ are all zero, and $\operatorname{Tr}(u^s) = 0$ for all $s > 0$ (Cor. 3). Conversely, if $\operatorname{Tr}(u^s) = 0$ for $1 \leq s \leq n$, then the $a_i$ are all zero since $K$ is of characteristic zero (IV, p. 72, Cor.), and $u$ is nilpotent (VII, p. 33).