Discriminants and Large Pólya Groups in Septic Number Fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911415837655040 |
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| author | Mahapatra, Nimish Kumar |
| author_facet | Mahapatra, Nimish Kumar |
| contents | We investigate a new family of cyclic septic fields $\{K_t\}_{t\in\mathbb{Z}}$ arising from the Hashimoto--Hoshi construction. For this family, we compute the discriminant explicitly and characterize their Pólya property under the condition that the polynomial $E(t) = t^{6} + 2t^{5} + 11t^{4} + t^{3} + 16t^{2} + 4t + 8$ takes fifth-power free values. We show that this family contains infinitely many non-Pólya fields for which the cardinality of the Pólya group is unbounded. We also establish that, assuming Bunyakovsky's conjecture for $E(t)$, this family contains infinitely many Pólya fields. We further show that, for any fixed positive integer $m$, there exist infinitely many blocks of $m$ consecutive fields in this family whose cardinality of the Pólya groups can be made arbitrarily large. Finally, we demonstrate that infinitely many fields in this family are non-monogenic with field index one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discriminants and Large Pólya Groups in Septic Number Fields Mahapatra, Nimish Kumar Number Theory 11R09, 11R29 We investigate a new family of cyclic septic fields $\{K_t\}_{t\in\mathbb{Z}}$ arising from the Hashimoto--Hoshi construction. For this family, we compute the discriminant explicitly and characterize their Pólya property under the condition that the polynomial $E(t) = t^{6} + 2t^{5} + 11t^{4} + t^{3} + 16t^{2} + 4t + 8$ takes fifth-power free values. We show that this family contains infinitely many non-Pólya fields for which the cardinality of the Pólya group is unbounded. We also establish that, assuming Bunyakovsky's conjecture for $E(t)$, this family contains infinitely many Pólya fields. We further show that, for any fixed positive integer $m$, there exist infinitely many blocks of $m$ consecutive fields in this family whose cardinality of the Pólya groups can be made arbitrarily large. Finally, we demonstrate that infinitely many fields in this family are non-monogenic with field index one. |
| title | Discriminants and Large Pólya Groups in Septic Number Fields |
| topic | Number Theory 11R09, 11R29 |
| url | https://arxiv.org/abs/2511.07954 |