Totally real bi-quadratic fields with large Pólya groups

Abstract

For an algebraic number field $K$ with ring of integers ${\mathcal{O}}_K$, an important subgroup of the ideal class group $C{l}_K$ is the {\it Pólya group}, denoted by $Po(K)$, which measures the failure of the ${\mathcal{O}}_K$-module $Int({\mathcal{O}}_K)$ of integer-valued polynomials on ${\mathcal{O}}_K$ from admitting a regular basis. In this paper, we prove that for any integer $n\ge 2$, there are infinitely many totally real bi-quadratic fields $K$ with $|Po(K)|=2^n$. In fact, we explicitly construct such an infinite family of number fields. This extends an infinite family of bi-quadratic fields with Pólya group $\mathbb{Z}/2\mathbb{Z}$ given by the authors in \cite{self-ja}. This also provides an infinite family of bi-quadratic fields with class numbers divisible by $2^n$.

arXiv comment7 pages