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