Let $V$ be a finite-dimensional vector space over $F$ with $char(F) = 0$ and $ T: V \rightarrow V $ a linear map. Suppose that $Tr(T^n)=0$ for all $n≥1.$ Show that $T$ nilpotent.
I have seen a proof that uses the Fitting's lemma. I believe there should be some straight-forward proof from scratch. Could you help me with some suggestions? Thanks so much.