Abstract:
Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. Let $\prod_q(K)$ be the product of all prime ideals of $\mathcal{O}_K$ with absolute norm $q.$ The Pólya group of a number field $K$ is the subgroup of the class group of $K$ generated by the classes of $\prod_q(K).$$K$ is a Pólya field if and only if the ideals $ \prod_{q}(K)$ are principal. In this paper, we follow the work that we have done in [S. EL Madrari, “On the Pólya fields of some real biquadratic fields”, Matematicki Vesnik,
online 05.09.2024]
where we studied the Pólya groups and fields in a particulare cases. Here, we will give the Pólya groups of $K=\mathbb{Q}(\sqrt{d}_1,\sqrt{d}_2)$ such that $d_1=lm_1$ and $d_2=lm_2$ are square-free integers with $l>1$ and $gcd(m_1,m_2)=1$ and the prime $2$ is not totally ramified in $K/\mathbb{Q}.$ And then, we characterize the Pólya fields of the real biquadratic fields $K.$
Keywords:Pólya fields, Pólya groups, real biquadratic fields, the first cohomology group of units, integer-valued polynomials
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