Totally real bi-quadratic fields with large Pólya groups

Abstract

For an algebraic number field K with ring of integers OK , an important subgroup of the ideal class group ClK is the Pólya group, denoted by P o(K), which measures the failure of the OK -module Int(OK) of integer-valued polynomials on OK from admitting a regular basis. In this paper, we prove that for any integer n ≥ 2, there are infinitely many totally real biquadratic fields K with |P o(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 Z/2Z given by the authors in [4]. This also provides an infinite family of bi-quadratic fields with class numbers divisible by 2 n .