Capitulation des 2-classes d’idéaux de certains corps biquadratiques dont le corps de genres diffère du 2-corps de classes de Hilbert

Abdelmalek Azizi and Ali Mouhib

Vol. 218 (2005), No. 1, 17–36

DOI: 10.2140/pjm.2005.218.17

Abstract

Let K = ℚ(,), where d₁ and d₂ are positive square-free integers such that (d₁,d₂) = 1. Let K₂⁽¹⁾ be the Hilbert 2-class field of K. Let K₂⁽²⁾ be the Hilbert 2-class field of K₂⁽¹⁾ and K^(∗) the genus field of K. We suppose that K₂⁽¹⁾≠K^(∗) and Gal(K₂⁽¹⁾∕K) ≃ ℤ∕2ℤ × ℤ∕2ℤ. We study the capitulation problem of the 2-ideal classes of K in the sub-extensions of K₂⁽¹⁾∕K and we determine the structure of Gal(K₂⁽²⁾).

Keywords

groupe de classes, capitulation, corps de classes de Hilbert

Mathematical Subject Classification 2000

Primary: 11R27, 11R37

Milestones

Received: 15 October 2002

Published: 1 January 2005