P\'olya-Ostrowski Group and Unit Index in Real Biquadratic Fields

TL;DR

This paper explores the Pólya-Ostrowski group in real biquadratic fields, establishing explicit relations with Hasse unit indices and refining bounds on ramified primes, advancing understanding of algebraic number field properties.

Contribution

It provides an explicit relation between Pólya groups and Hasse unit indices in real biquadratic fields, enhancing previous theoretical results.

Findings

Relation between Pólya groups and Hasse unit indices established

Refined upper bound on ramified primes in Pólya real biquadratic fields

Improved understanding of the structure of Pólya groups in specific number fields

Abstract

The P\'olya-Ostrowski group of a Galois number field $K$, is the subgroup $Po(K)$ of the ideal class group $Cl(K)$ of $K$ generated by the classes of all the strongly ambiguous ideals of $K$. The number field $K$ is called a P\'olya field, whenever $Po(K)$ is trivial. In this paper, using some results of Bennett Setzer \cite{Bennett} and Zantema \cite{Zantema}, we give an explicit relation between the order of P\'olya groups and the Hasse unit indices in real biquadratic fields. As an application, we refine Zantema's upper bound on the number of ramified primes in P\'olya real biquadratic fields.