Abstract

Let $\mathcal{O}_K$ be the ring of integers of a number field $K$, and let $\mathrm{Int}(\mathcal{O}_K)=\{R\in K[X] \mid R(\mathcal{O}_K)\subset \mathcal{O}_K\}$. The Pólya group is the group generated by the classes of the products of the prime ideals with the same norm. The Pólya group $\mathcal{P}_O(K)$ is trivial if and only if the $\mathcal{O}_K$-module $\mathrm{Int}(\mathcal{O}_K)$ has a regular basis if and only if $K$ is a Pólya field. In this paper, we give the structure of the first cohomology group of units of the real biquadratic number fields $K=\mathbb{Q}(\sqrt{d}_1, \sqrt{d}_2)$, where $d_1>1$ and $d_2>1$ are two square-free integers with $(d_1,d_2)=1$ and the prime $2$ is not totally ramified in $K/\mathbb{Q}$. We then determine the Pólya groups and the Pólya fields of $K$.

Keywords: Pólya fields; Pólya groups; real biquadratic fields; the first cohomology group of units; integer-valued polynomials.

MSC: 11R04, 11R16, 11R27, 13F20

DOI: 10.57016/MV-N58UKM54

Pages:  1--16