Brief communications

Fields over which matrices can be represented as the sum of potent and nilpotent matrices

A. N. Abyzov^ab, D. T. Tapkin<sup>ab

a</sup> Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia
^b Emperor Alexander I St. Petersburg State Transport University, 9 Moskovsky Ave., Saint Petersburg, 190031 Russia

Abstract: We study fields over which every matrix can be represented as a sum of two potent matrices and a nilpotent matrix. In particular, it is shown that over a field $P$ every matrix can be represented as a sum of two idempotent matrices and a nilpotent matrix exactly when either $P\cong \mathbb{F}_2,$ or $P\cong \mathbb{F}_3$.

Keywords: $q$-potent, finite field, matrices over finite fields.

UDC: 512.552

Received: 17.09.2025
Revised: 17.09.2025
Accepted: 26.09.2025

DOI: 10.26907/0021-3446-2025-10-78-82