Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be an endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if it satisfies the following two conditions:
(a) there exist natural integers $c_1, \ldots, c_n$ such that $\chi_u = \prod_{\ell=1}^{n} \left(X^\ell - 1\right)^{c_\ell}$;
(b) there exists $N$ such that $u^N = \operatorname{Id}_E$.