Rank of Pólya Groups in Lecacheux Parametric Family of Quintic Fields

Nimish Kumar Mahapatra; Prem Prakash Pandey

Rank of Pólya Groups in Lecacheux Parametric Family of Quintic Fields

Number Theory 2026-07-01 v1

Authors: Nimish Kumar Mahapatra , Prem Prakash Pandey

Abstract

In this article, we study the P\'olya group of a new family of quintic fields, namely Lecacheux quintic fields. We show that the associated P\'olya groups can be arbitrarily large elementary abelian $5$ -groups. Using density arguments, we prove that for every positive integer $k$ , the set of odd integers $s$ such that the $5-$ rank of the P\'olya group of the corresponding Lecacheux quintic field is at least $k$ has a positive density. Combining this with a result of Golod and Shafarevich, we see that for a positive proportion of $s$ , the corresponding Lecacheux quintic fields admit an infinte $5-$ class field tower. We also establish an upper bound for the P\'olya numbers of these fields in terms of the orders of their corresponding P\'olya groups. In addition, we prove that several fields in this family are non-monogenic despite having index one.

Keywords

Pólya groups quintic fields Lecacheux fields elementary abelian groups 5-rank class field tower non-monogenic

Cite

@article{arxiv.2607.00675,
  title  = {Rank of Pólya Groups in Lecacheux Parametric Family of Quintic Fields},
  author = {Nimish Kumar Mahapatra and Prem Prakash Pandey},
  journal= {arXiv preprint arXiv:2607.00675},
  year   = {2026}
}