Large P\'{o}lya groups in simplest cubic fields and consecutive bi-qaudratic fields

Abstract

The P\'{o}lya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of {\it consecutiveness} of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large P\'{o}lya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields.