Companion matrices as sums of $p$-potent and nilpotent matrices

Abstract

We prove that, over a field $\mathbb{F}$ of odd characteristic $p$, a companion matrix $C$ is the sum of $E$ and $N$, with $E$ $p$-potent (i.e. $E^p=E$,) and $N$ nilpotent, if and only if the trace of $C$ is an integer multiple of unity of $\mathbb{F}$.