Discriminants and Large P\'olya Groups in Septic Number Fields

TL;DR

This paper studies a family of cyclic septic number fields, analyzing their discriminants, Pólya properties, and monogenicity, revealing infinitely many with unbounded Pólya groups and some with specific algebraic features.

Contribution

It introduces a new family of septic fields from the Hashimoto–Hoshi construction and characterizes their Pólya properties and discriminants, including conditions for infinitely many Pólya fields.

Findings

Infinitely many non-Pólya fields with unbounded Pólya groups.

Conditional existence of infinitely many Pólya fields assuming Bunyakovsky's conjecture.

Existence of infinitely many fields with arbitrarily large Pólya groups and non-monogenic fields with index one.

Abstract

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 P\'olya 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-P\'olya fields for which the cardinality of the P\'olya group is unbounded. We also establish that, assuming Bunyakovsky's conjecture for $E(t)$, this family contains infinitely many P\'olya 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 P\'olya groups can be made arbitrarily large. Finally, we demonstrate that infinitely many fields in this family are non-monogenic with field index one.