Non-Pólya bi-quadratic fields with an Euclidean ideal class

Abstract

In this paper, we prove the existence of three pairwise distinct families of totally real bi-quadratic fields, each having Pólya group isomorphic to Z/2Z. This extends the previously known families of number fields considered by Heidaryan and Rajaei. Our results also establish that under mild assumptions, the possibly infinite families of bi-quadratic fields having a non-principal Euclidean ideal class, considered by Chattopadhyay and Muthukrishnan, fail to be Pólya fields.