= Nilpotence from vanishing traces of powers
{title2=$\operatorname{tr}(A^k)=0\ (1\le k\le n)\ \Longrightarrow\ A^n=0$}
For a complex $n$ by $n$ <matrix>, triangularization gives $\operatorname{tr}(A^k)=\sum_j\lambda_j^k$. The <zero power sums force a finite complex multiset to vanish> criterion makes every diagonal eigenvalue zero. A strictly upper-triangular $n$ by $n$ matrix has zero $n$th power: every nonzero product entry would require $n$ strictly increasing index steps. <Similarity> then gives $A^n=0$.
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