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

TL;DR

This paper investigates the divisibility properties of the multiplicative order modulo monic irreducible polynomials over finite fields, establishing conditions for the existence of their densities and providing formulas under certain assumptions.

Contribution

It introduces the concept of density for sets of polynomials with divisible multiplicative order and derives explicit formulas for these densities under specific conditions.

Findings

Sets have a Dirichlet density.

Existence of density depends on the notion of density used.

Provided formulas for density values under assumptions.

Abstract

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ 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.