Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields

Abstract

The Pólya group $Po(K)$ of an algebraic number field $K$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of the same norm. If $Po(K)$ is trivial, then the number field $K$ is said to be a Pólya field. In this article, we furnish three families $\mathbb{Q}(\sqrt{p},\sqrt{qrs})$, $\mathbb{Q}(\sqrt{2p},\sqrt{qrs})$ and $\mathbb{Q}(\sqrt{2p},\sqrt{2qrs})$ of bi-quadratic Pólya fields $K$ involving prime numbers $p,q,r$ and $s$ that satisfy certain quadratic residue conditions. It is worthwhile to note that in each of the fields, exactly five primes ramify in $K/\mathbb{Q}$ and this is the maximum possible number of ramified primes in a Pólya field over $\mathbb{Q}$. Towards the end of the paper, we discuss about large Pólya groups of the compositums of Shank's cubic fields and Lehmer's quintic fields and prove that there are infinitely many such fields with index $1$.