A brief survey of recent results on Pólya groups

Abstract

The Pólya group of an algebraic number field is a particular subgroup of the ideal class group. This article provides an overview of recent results on Pólya groups of number fields, their connection with the ring of integer-valued polynomials and touches upon some results on number fields having large Pólya groups. For the sake of completeness, we have included the proof of Zantema's theorem which laid the foundation to determine the Pólya groups of many finite Galois extensions over Q. Towards the end of the article, we provide an elementary proof of a weaker version of a recent result of Cherubini et al.