Divisibility of the multiplicative order modulo monic irreducible polynomials over finite fields

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 $F_q\left(T\right)$ 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.