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

TL;DR

This paper uses advanced number theory techniques to show that, assuming GRH, certain primes lead to factorizations in monoid rings, revealing new properties of specific monoids and extending prior observations.

Contribution

It proves that under GRH, the set of primes with particular factorizations in monoid rings has positive density, demonstrating that a specific monoid is not a Matsuda monoid of any positive type.

Findings

Set of primes with factorizations has positive density

Confirmed non-Matsuda monoid property for M

Reproduced known factorizations for p=2,3

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.