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

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 Pólya 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 $C{l}_K$, known as the {\it Pólya group of $K$}, that measures the failure of $K$ from being a Pólya field. In this paper, we prove the existence of three pairwise distinct totally real bi-quadratic fields, each having Pólya 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 Pólya fields.

arXiv comment12 pages