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

TL;DR

Let $K=\mathbb{F}_q(T)$ and $R_q(a,d)$ be the set of monic irreducible $P\in \mathbb{F}_q[T]$ with $d$ dividing the multiplicative order of $a$ modulo $P$, i.e., $d\mid\operatorname{ord}_P(a)$. The paper develops a function-field analogue of the classical order-divisibility problem by expressing $R_q(a,d,N)$ as counts of primes that split completely in Kummer extensions $K(\zeta_n,a^{1/d})$, and then applies a Chebotarev-type theorem for global function fields to obtain asymptotics of the form $R_q(a,d,N) = (q^N/N)\,\delta_q(a,d,N) + O(q^{N/2}/N)$. It is shown that $R_q(a,d)$ typically lacks $d_1$-density but possesses a robust $d_3$-density $\delta_q(a,d)$, with $\delta_q(a,d,N)$ converging to $\delta_q(a,d)$; under a natural irreducibility assumption on Kummer extensions, a closed Euler-product for $\delta_q(a,d)$ is derived. The results generalize and connect to Artin-type primitive-root phenomena in function fields, providing explicit density formulas and a structural framework via constant-field extensions and Kummer theory. The work yields both qualitative density results (existence/nonexistence of certain densities) and quantitative, computable expressions for the density, enriching the understanding of multiplicative orders in polynomial residue rings over finite fields.

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.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Lemma 1
• proof