Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.
Let $u$ and $v$ be two endomorphisms of $E$ such that, for all $k \in \mathbb{N}, \operatorname{Tr}\left(u^k\right) = \operatorname{Tr}\left(v^k\right)$. Show that $u$ and $v$ have the same characteristic polynomial.