Finite fields whose members are the sum of a potent and a 5-potent
Number Theory
2026-01-13 v2 Commutative Algebra
Rings and Algebras
Abstract
We show that there are only finitely many finite fields whose members are the sum of an -potent element and a -potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the affirmative the conjecture in \cite{Cohen} concerning all finite fields satisfying this condition. Furthermore, we obtain several elementary results for General problem, proving that the number of finite fields satisfying general condition is also finite.
Keywords
Cite
@article{arxiv.2512.16942,
title = {Finite fields whose members are the sum of a potent and a 5-potent},
author = {Juncheng Zhou and Peter V. Danchev and Hongfeng Wu},
journal= {arXiv preprint arXiv:2512.16942},
year = {2026}
}