Some bi-quadratic P\'{o}lya fields and large P\'{o}lya groups of compositum of simplest cubic and quintic fields

Abstract

The P\'{o}lya 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 P\'{o}lya 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 P\'{o}lya 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 P\'{o}lya field over $\mathbb{Q}$. Towards the end of the paper, we discuss about large P\'{o}lya 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 $1$.