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

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.