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

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 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 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 , this family contains infinitely many P\'olya fields. We further show that, for any fixed positive integer , there exist infinitely many blocks of 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…
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
