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.