An application of a generalization of Artin's primitive root conjecture in the theory of monoid rings

Abstract

Using techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman, et. al., under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate GRH, for any (rational) prime $p$ the set $E(p) = \{ q \text{ prime } \, | \, X^q - 1 \text{ factors in } \mathbb{F}_p[X;M] \}$, where $M = \langle 2, 3 \rangle = \mathbb{N}_0 \setminus \{ 1 \}$, contains a subset with positive natural density. In particular $E(p) \ne \varnothing$. This proves that $M$ is not a so-called ``Matsuda monoid'' of any positive type. For $p = 2, 3$ this was observed by Kulosman, who provided factorizations of $X^7-1$ and $X^{11} - 1$ in $\mathbb{F}_2[X; M]$ and $\mathbb{F}_3[X;M]$, respectively. Our results explain and reproduce both of these factorizations, as well.