On P\'olya groups of some non-Galois number fields

TL;DR

This paper proves conjectures about Pólya groups in certain non-Galois number fields, extending known results to broader classes and establishing new equivalences for prime degree fields.

Contribution

It confirms two conjectures for specific non-Galois fields and generalizes results to D_n-fields with even n, also relating pre-Pólya and Pólya groups in prime degree fields.

Findings

Confirmed conjectures for S_4 and D_4 fields.

Extended results to D_n-fields with even n.

Established that pre-Pólya and Pólya groups coincide in prime degree fields.

Abstract

We prove two conjectures proposed by Chabert and Halberstadt concerning P\'olya groups of $S_4$-fields and $D_4$-fields. More generally, the latter will be proved for $D_n$-fields with $n \geq 4$ an even integer. Further, generalizing a result of Zantema, we also prove that the pre-P\'olya group of a non-Galois field of a prime degree, e.g. an $A_5$-field, coincides with its P\'olya group.