Pólya groups of the imaginary bicyclic biquadratic number fields

Abstract

Let OK and CK be respectively the ring of integers and the class group of a number field K. For each integer q≥2, denote by ∏q(K) the product of all the maximal ideals of OK with norm q, if these ideals do not exist we set ∏q(K)=OK. The Pólya group of K is the subgroup of CK generated by the classes of the ideals ∏q(K), and K is called a Pólya field if the module of integer-valued polynomials over OK has a regular basis. In this paper, we determine Pólya group of any imaginary bicyclic biquadratic number field, and thus we deduce all the imaginary bicyclic biquadratic Pólya fields.