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

TL;DR

This paper constructs specific families of bi-quadratic Pólya fields with maximal ramification and explores large Pólya groups in compositums of cubic and quintic fields, demonstrating infinite instances with index 1.

Contribution

It introduces new families of bi-quadratic Pólya fields with maximal ramification and analyzes large Pólya groups in compositums of special cubic and quintic fields, showing infinite examples with index 1.

Findings

Constructed three families of bi-quadratic Pólya fields with five ramified primes.

Proved existence of infinitely many fields with large Pólya groups and index 1.

Identified maximal ramification in Pólya fields over b5Q.

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 $1$.