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Pólya groups of the imaginary bicyclic biquadratic number fields. (English) Zbl 1428.11185

Summary: Let \(\mathcal{O}_K\) and \(\operatorname{C}_K\) be respectively the ring of integers and the class group of a number field \(K\). For each integer \(q \geq 2\), denote by \(\prod_q(K)\) the product of all the maximal ideals of \(\mathcal{O}_K\) with norm \(q\), if these ideals do not exist we set \(\prod_q(K) = \mathcal{O}_K\). The Pólya group of \(K\) is the subgroup of \(\operatorname{C}_K\) generated by the classes of the ideals \(\prod_q(K)\), and \(K\) is called a Pólya field if the module of integer-valued polynomials over \(\mathcal{O}_K\) has a regular basis. In this paper, we determine Pólya group of any imaginary bicyclic biquadratic number field, and thus we deduce all the imaginary bicyclic biquadratic Pólya fields.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R16 Cubic and quartic extensions
11R27 Units and factorization
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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