On the monogenity of pure number fields: application to the existence of canonical number systems
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| Format: | Preprint |
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2024
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| _version_ | 1866917002231152640 |
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| author | Yakkou, Hamid Ben Boudine, Brahim Tiebekabe, Pagdame |
| author_facet | Yakkou, Hamid Ben Boudine, Brahim Tiebekabe, Pagdame |
| contents | Let $m$ be a rational integer with $m \neq 0, \pm 1$, and consider the pure number field $K = \mathbb{Q}(\sqrt[n]{m})$ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus exclusively on the case where $m$ is square-free. For every integer $n \ge 4$, the monogenity of number fields of degree $n$ is not completely characterized. For example, the monogenity of the pure quartic field $\mathbb{Q}(\sqrt[4]{m})$ is not yet fully described, even when $m$ is square-free (see the recent 2024 paper \cite{Nyul} by Arnóczki and Nyul). In this paper, based on a classical theorem of Ore concerning prime ideal decomposition in number fields \cite{MN92, O}, we study the monogenity of $K$ without assuming $m$ to be square-free. As an application, we present several examples related to canonical number systems (CNS). In particular, we observe that our results extend some of those presented in \cite{BFC, BF, HNHCNS}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_00819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the monogenity of pure number fields: application to the existence of canonical number systems Yakkou, Hamid Ben Boudine, Brahim Tiebekabe, Pagdame Number Theory 11A63, 11R04, 11A63, 11R04, 11R16, 11R21, 11Y40 F.2.2 Let $m$ be a rational integer with $m \neq 0, \pm 1$, and consider the pure number field $K = \mathbb{Q}(\sqrt[n]{m})$ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus exclusively on the case where $m$ is square-free. For every integer $n \ge 4$, the monogenity of number fields of degree $n$ is not completely characterized. For example, the monogenity of the pure quartic field $\mathbb{Q}(\sqrt[4]{m})$ is not yet fully described, even when $m$ is square-free (see the recent 2024 paper \cite{Nyul} by Arnóczki and Nyul). In this paper, based on a classical theorem of Ore concerning prime ideal decomposition in number fields \cite{MN92, O}, we study the monogenity of $K$ without assuming $m$ to be square-free. As an application, we present several examples related to canonical number systems (CNS). In particular, we observe that our results extend some of those presented in \cite{BFC, BF, HNHCNS}. |
| title | On the monogenity of pure number fields: application to the existence of canonical number systems |
| topic | Number Theory 11A63, 11R04, 11A63, 11R04, 11R16, 11R21, 11Y40 F.2.2 |
| url | https://arxiv.org/abs/2407.00819 |