Non-P\'olya Fields with Large P\'olya Groups Arising from Lehmer Quintics

TL;DR

This paper constructs a new family of quintic non-Pólya fields with large Pólya groups, analyzes their Pólya numbers, and shows they are non-monogenic fields with specific properties, expanding understanding of algebraic number fields.

Contribution

It introduces a novel family of non-Pólya quintic fields with large Pólya groups and investigates their Pólya numbers and monogenicity properties.

Findings

Pólya numbers are at most five times the Pólya group size.

Constructed fields are non-monogenic with field index one.

Established bounds on Pólya numbers for these fields.

Abstract

In this article we construct a new family of quintic non-P\'olya fields with large P\'olya groups. We study the upper bound of P\'olya numbers of such fields and show that the P\'olya numbers never exceed five times the size of its P\'olya group. Finally we show that such non-P\'olya fields are non-monogenic fields of field index one.