An Infinite Family of Septic Number Fields with Large Pólya Groups
Nimish Kumar Mahapatra
TL;DR
The paper classifies the Pólya property in the Hashimoto--Hoshi family of cyclic septic fields $K_t$ by linking the Pólya group to the prime factorization of $E(t)=t^{6}+2t^{5}+11t^{4}+t^{3}+16t^{2}+4t+8$. Under the assumption that $E(t)$ is fifth-power free, the Pólya group is $(\\mathbb{Z}/7\\mathbb{Z})^{\omega(E(t))-2}$ for even $t$ and $(\Z/7\Z)^{\omega(E(t))-1}$ for odd $t$, yielding infinitely many non-Pólya fields with unbounded $7$-rank and, conditionally on Bunyakovsky for $E$, infinitely many Pólya fields. The authors further show that for any fixed $m$, there exist infinitely many blocks of $m$ consecutive $t$ with uniformly large Pólya groups, and prove non-monogenicity with index $1$ for infinitely many odd $t$ coprime to $15$. The results connect the distribution of values of $E(t)$ to both Pólya properties and monogeneity in a high-degree, explicitly constructed family, and they discuss conditional versus unconditional prospects for establishing infinite Pólya families.
Abstract
We investigate a new family of cyclic septic fields $\{K_t\}_{t\in\mathbb{Z}}$ arising from the Hashimoto--Hoshi construction 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.