= Classification of integral matrices satisfying the third cyclotomic polynomial
An integral matrix satisfying $A^2+A+I=0$ makes $\mathbb Z^n$ a module over the <Eisenstein integers> by letting the primitive cube root act as $A$. This module is finitely generated and torsion-free, hence free. Since the Eisenstein integers have integer rank two, such matrices exist only for $n=2m$; for that dimension every one is integrally conjugate to
$$
\begin{pmatrix}0&-1\\1&-1\end{pmatrix}^{\oplus m}.
$$
Back to article page
