On integral bases and monogenity of pure octic number fields with non-square free parameters
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| Format: | Preprint |
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2022
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| _version_ | 1866911776630636544 |
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| author | Fadil, Lhoussain El Gaál, István |
| author_facet | Fadil, Lhoussain El Gaál, István |
| contents | In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $α$ of a monic irreducible polynomial $f(x)=x^8-m\in\mathbf Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We start by calculating an integral basis of $\mathbf Z_K$, the ring of integers of $K$. In particular, we characterize when $\mathbf Z_K=\mathbf Z[α]$. We give sufficient conditions on $m$, which guarantee that $K$ is not monogenic. We finish the paper by investigating the case when $m=a^u$, $u\in\{1,3,5,7\}$ and $a\neq \mp 1$ is a square free rational integer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2202_04417 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On integral bases and monogenity of pure octic number fields with non-square free parameters Fadil, Lhoussain El Gaál, István Number Theory 11R04, 11R16, 11R21 In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $α$ of a monic irreducible polynomial $f(x)=x^8-m\in\mathbf Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We start by calculating an integral basis of $\mathbf Z_K$, the ring of integers of $K$. In particular, we characterize when $\mathbf Z_K=\mathbf Z[α]$. We give sufficient conditions on $m$, which guarantee that $K$ is not monogenic. We finish the paper by investigating the case when $m=a^u$, $u\in\{1,3,5,7\}$ and $a\neq \mp 1$ is a square free rational integer. |
| title | On integral bases and monogenity of pure octic number fields with non-square free parameters |
| topic | Number Theory 11R04, 11R16, 11R21 |
| url | https://arxiv.org/abs/2202.04417 |