Totally real bi-quadratic fields with large Pólya groups

For an algebraic number field K with ring of integers $$\mathcal {O}_{K}$$ , important subgroup the ideal class group $$Cl_{K}$$ is Pólya group, denoted by Po(K), which measures failure -module $$Int(\mathcal {O}_{K})$$ integer-valued polynomials on 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 $$Po(K) \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{n}$$ . fact, explicitly construct such infinite family fields. This also provides groups 2-ranks at least n.