Totally real bi-quadratic fields with large Pólya groups.

For an algebraic number field K with ring of integers O K , an important subgroup of the ideal class group C l K is the Pólya group, denoted by Po(K), which measures the failure of the O K -module I n t (O K) of integer-valued polynomials on O K from admitting a regular basis. In this paper, we prove that for any integer n ≥ 2 , there are infinitely many totally real bi-quadratic fields K with P o (K) ≃ (Z / 2 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. [ABSTRACT FROM AUTHOR]