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