Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields

Joaquim Cera Da Conceição

doi:10.1016/j.jnt.2025.03.003

paper

Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields

arXiv:2412.09107 · doi:10.1016/j.jnt.2025.03.003

Abstract

We consider the set of monic irreducible polynomials $P$ over a finite field $F_{q}$ such that the multiplicative order modulo $P$ of some a in $F_{q} (T)$ is divisible by a fixed positive integer $d$ . Call $R_{q} (a, d)$ this set. We show the existence or non-existence of the density of $R_{q} (a, d)$ for three distinct notions of density. In particular, the sets $R_{q} (a, d)$ have a Dirichlet density. Under some assumptions, we prove simple formulas for the density values.

16 pages