Pólya-Ostrowski Group and Unit Index in Real Biquadratic Fields

TL;DR

The paper establishes a uniform, explicit relation between the order of the Pólya-Ostrowski group $|Po(K)|$ of a real biquadratic field and its Hasse unit index, building on Bennett Setzer and Zantema. The main result expresses $|Po(K)|$ in terms of the Hasse unit index $[U_K:U_{k_1}U_{k_2}U_{k_3}]$, the ramification product $\prod_{p|d_K} e_p(K/\mathbb{Q})$, and $t$ (the number of quadratic subfields with negative-norm fundamental units), with a refined case distinction tied to the behavior of $2$ and unit norms; notably, $t=2$ or $t=3$ yields equal Po(K) orders. As an application, the authors refine Zantema's ramification bound for Pólya real biquadratic fields, deriving tighter limits on the number of ramified primes depending on $t$ and unit data. Overall, the work provides a unified, unit-index–driven framework to compute $Po(K)$ and to identify Pólya fields in the real biquadratic setting, with implications for ramification constraints.

Abstract

The Pólya-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 Pólya 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 Pólya 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 Pólya real biquadratic fields.

Theorems & Definitions (10)

• Proposition 1.1
• Proposition 2.1
• Theorem 2.2
• proof
• Proposition 2.3
• Example 2.4
• Remark 2.5
• Proposition 2.6
• Corollary 2.7
• proof