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Capitulation of 2-ideal classes of \({\mathbb Q}(\sqrt{2},\sqrt{d})\) where \(d\) is a square-free natural integer. (Capitulation des 2-classes d’idéaux de \({\mathbb Q}(\sqrt{2},\sqrt{d})\) où \(d\) est un entier naturel sans facteurs carrés.) (French) Zbl 1077.11078

Let \(K=\mathbb{Q}(\sqrt{ 2}, \sqrt{d}) \) be a biquadratic number field, where \(d\) is a square-free natural number. Let \(K_{2}^{(1)}\) (resp. \(K_{2}^{(2)}\)) be the Hilbert \(2\)-class field of \(K\) (resp. \(K_{2}^{(1)}\)). The authors give a necessary and sufficient condition for \(K\), in terms of the Legendre symbol, to have Gal\((K_{2}^{(1)}/K)\) isomorphic to \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\). In this case, using results of H. Kisilevsky [J. Number Theory 8, 271–279 (1976; Zbl 0334.12019)], they determine the structure of Gal\((K_{2}^{(2)} /K)\) and show that Gal\((K_{2}^{(2)}/K)\) is either abelian, dihedral or quaternionic.

MSC:

11R27 Units and factorization
11R37 Class field theory

Citations:

Zbl 0334.12019
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