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 $Cl_{K}$ is the 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 $Po(K) \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{n}$. In fact, we explicitly construct such an infinite family of number fields. This also provides an infinite family of bi-quadratic fields with ideal class groups of 2-ranks at least n.