What can we say about the Pólya Group
of a Bicyclic Biquadratic Number Field ?
Jean-Luc Chabert
LAMFA CNRS-UMR 7352, Université de Picardie, 80039 Amiens, France.
Pages 70-82 | Received 21 October 2022, Accepted 13 January 2023, Published 16 June 2023
Abstract
The Pólya group \mathcal{P} o(K) of a finite Galois extension K of \mathbb{Q} is the subgroup of the class group of K formed by the strong ambiguous classes of K. In this paper, we state a general formula which gives the order of \mathcal{P} o(K) when K is a bicyclic biquadratic number field by means of classical indices, namely, the unit index of K, the number of ramified primes, and the number of fundamental units with norm 1 of the quadratic subfields of K. Then we study separately the imaginary case and the real case.
Keywords: Pólya group, Pólya field, Biquadratic number field, Ambiguous ideal.
MSC numbers: Primary 11R20, 11R29; Secondary 13F20.
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