On the monogenity of totally complex pure octic fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909106474844160 |
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| author | Gaál, István |
| author_facet | Gaál, István |
| contents | Let $0,1\ne m\in Z$ and $α=\sqrt[8]{m}$. According to the results of I. Gaál and L. El Fadil, $α$ generates a power integral basis in $K=Q(α)$, if and only if $m$ is square-free and $m\not\equiv 1\;(\bmod\; 4)$. In the present paper we consider totally complex pure octic fields, that is the case $m<0$, with $m$ satisfiying the above property. In this case $(1,α,α^2,\ldots,α^7)$ is an integral basis. Our purpose is to investigate whether $K$ admits any other generators of power integral bases, inequivalent to $α$. We present an efficient method to calculate generators of power integral bases in this type of fields with coefficients $<10^{200}$ in the above integral basis. We report on the results of our calculation for this type of fields with $0>m>-5000$, which yields 2024 fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09293 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the monogenity of totally complex pure octic fields Gaál, István Number Theory 11Y50, 11R04 Let $0,1\ne m\in Z$ and $α=\sqrt[8]{m}$. According to the results of I. Gaál and L. El Fadil, $α$ generates a power integral basis in $K=Q(α)$, if and only if $m$ is square-free and $m\not\equiv 1\;(\bmod\; 4)$. In the present paper we consider totally complex pure octic fields, that is the case $m<0$, with $m$ satisfiying the above property. In this case $(1,α,α^2,\ldots,α^7)$ is an integral basis. Our purpose is to investigate whether $K$ admits any other generators of power integral bases, inequivalent to $α$. We present an efficient method to calculate generators of power integral bases in this type of fields with coefficients $<10^{200}$ in the above integral basis. We report on the results of our calculation for this type of fields with $0>m>-5000$, which yields 2024 fields. |
| title | On the monogenity of totally complex pure octic fields |
| topic | Number Theory 11Y50, 11R04 |
| url | https://arxiv.org/abs/2402.09293 |