A brief survey of recent results on P\'{o}lya groups

Abstract

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