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

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.