## 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

### Keywords:

class group; capitulation; Hilbert class field

Zbl 0334.12019
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