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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 particularwhen $R$ is the ring of integers of an algebraic number field. An algebraicnumber field $K$ with ring of integers $\mathcal{O}_{K}$ is said to be aP\'{o}lya field if the $\mathcal{O}_{K}$-module of integer-valued polynomialson $K$ admits a regular basis. Associated to $K$ is a subgroup $Po(K)$ of theideal class group $Cl_{K}$, known as the {\it P\'{o}lya group of $K$}, thatmeasures the failure of $K$ from being a P\'{o}lya field. In this paper, weprove the existence of three pairwise distinct totally real bi-quadraticfields, each having P\'{o}lya group isomorphic to $\mathbb{Z}/2\mathbb{Z}$.This extends the previously known families of number fields considered byHeidaryan and Rajaei in \cite{rajaei-jnt} and \cite{rajaei}. Our results alsoestablish that under mild assumptions, the possibly infinite families ofbi-quadratic fields having a non-principal Euclidean ideal class, considered in\cite{self-jnt}, fail to be P\'{o}lya fields.

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