An application of a generalization of Artin's primitive root conjecture in the theory of monoid rings
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910958024130560 |
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| author | Daileda, Ryan C. |
| author_facet | Daileda, Ryan C. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_09080 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | An application of a generalization of Artin's primitive root conjecture in the theory of monoid rings Daileda, Ryan C. Number Theory 11Z05 (Primary), 12F10, 20M25 (Secondary) 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. |
| title | An application of a generalization of Artin's primitive root conjecture in the theory of monoid rings |
| topic | Number Theory 11Z05 (Primary), 12F10, 20M25 (Secondary) |
| url | https://arxiv.org/abs/2112.09080 |