On the capitulation of the $2$-ideal classes of the field Q(\sqrt{pq_1q_2}, i) of type (2, 2, 2)

Abdelmalek Azizi; Abdelkader Zekhnini; Mohammed Taous

doi:10.5269/bspm.v38i4.36793

Abdelmalek Azizi Mohammed First University Sciences Faculty Mathematics Department http://orcid.org/0000-0002-0634-1995
Abdelkader Zekhnini Mohammed First university Pluridisciplinary Faculty of Nador http://orcid.org/0000-0002-0900-4708
Mohammed Taous Moulay Ismail University Sciences and Techniques Faculty Mathematics Department http://orcid.org/0000-0003-3366-5124

DOI: https://doi.org/10.5269/bspm.v38i4.36793

Keywords: absolute genus fields, fundamental systems of units, $2$-class group, capitulation, quadratic fields, biquadratic fields, multiquadratic CM-fields

Abstract

We study the capitulation of the 2-ideal classes of the field k =Q(\sqrt{p_1p_2q}, \sqrt{-1}), where p_1\equiv p_2\equiv-q\equiv1 \pmod 4 are different primes, in its three quadratic extensions contained in its absolute genus field k^{*} whenever the 2-class group of $\kk$ is of type $(2, 2, 2)$.