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arXiv · 2412.09107

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

Joaquim Cera Da Conceição
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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.

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Joaquim Cera Da Conceição. 2024-12-12. Divisibility of the Multiplicative Order Modulo Monic Irreducible Polynomials Over Finite Fields. https://doi.org/10.1016/j.jnt.2025.03.003

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