Non-P\'{o}lya bi-quadratic fields with an Euclidean ideal class

TL;DR

This paper constructs specific totally real bi-quadratic fields with a Pólya group of order two, extending known families and showing that some bi-quadratic fields with Euclidean ideal classes are not Pólya fields.

Contribution

It proves the existence of three distinct totally real bi-quadratic fields with Pólya group isomorphic to Z/2Z, expanding previous classifications and linking Euclidean ideal classes to Pólya field properties.

Findings

Existence of three bi-quadratic fields with Pólya group Z/2Z

Extension of known families of Pólya fields

Bi-quadratic fields with Euclidean ideal classes are not necessarily Pólya fields

Abstract

For an integral domain $R$, the {\it ring of integer-valued polynomials} over $R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$. An interesting case to study is when $R$ is a Dedekind domain, in particular when $R$ is the ring of integers of an algebraic number field. An algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$ is said to be a P\'{o}lya field if the $\mathcal{O}_{K}$-module of integer-valued polynomials on $K$ admits a regular basis. Associated to $K$ is a subgroup $Po(K)$ of the ideal class group $Cl_{K}$, known as the {\it P\'{o}lya group of $K$}, that measures the failure of $K$ from being a P\'{o}lya field. In this paper, we prove the existence of three pairwise distinct totally real bi-quadratic fields, each having P\'{o}lya group isomorphic to $\mathbb{Z}/2\mathbb{Z}$. This extends the previously known families of number fields considered by Heidaryan and Rajaei in \cite{rajaei-jnt} and \cite{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 in \cite{self-jnt}, fail to be P\'{o}lya fields.