An Infinite Family of Septic Number Fields with Large Pólya Groups

2025/11/11 by Mahapatra, Nimish Kumar
#11R09 #11R29 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.07954

Abstract

We investigate a new family of cyclic septic fields \K_t\_t∈ℤ arising from the Hashimoto--Hoshi construction and characterize their Pólya property under the condition that the polynomial E(t) = t⁶ + 2t⁵ + 11t⁴ + t³ + 16t² + 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.

Citations

• Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields
• Non-Pólya Fields with Large Pólya Groups Arising from Lehmer Quintics
• Totally real bi-quadratic fields with large Pólya groups
• A geometric approach to large class groups: a survey
• Non monogénéité de l'anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5

Cited by

• Number Fields With Large Pólya Groups

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• Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields
• On Pólya groups of some non-Galois number fields