Moroccan Journal of Algebra and Geometry with Applications

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What can we say about the Pólya Group
of a Bicyclic Biquadratic Number Field ?

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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