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Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 81–97 | Cite as

Capitulation in the Absolutely Abelian Extensions of some Number Fields II

  • Abdelmalek Azizi
  • Abdelkader ZekhniniEmail author
  • Mohammed Taous
Article
  • 74 Downloads

Abstract

We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields \(\mathbb {k} =\mathbb {Q}(\sqrt {pq_{1}q_{2}}, i)\), where \(i=\sqrt {-1}\) and q 1q 2≡−p≡−1 (mod 4) are different primes. For each of the three quadratic extensions \(\mathbb {K}/\mathbb {k}\) inside the absolute genus field 𝕜 (∗) of 𝕜, we compute the capitulation kernel of \(\mathbb {K}/\mathbb {k}\). Then we deduce that each strongly ambiguous class of \(\mathbb {k}/\mathbb {Q}(i)\) capitulates already in 𝕜 (∗).

Keywords

Absolute and relative genus fields Fundamental systems of units 2-class group Capitulation Quadratic fields Biquadratic fields Multiquadratic CM-fields 

Mathematics Subject Classification (2010)

11R11 11R16 11R20 11R27 11R29 

Notes

Acknowledgments

We would like to thank the referee for his/her precious remarks and suggestions.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  • Abdelmalek Azizi
    • 1
  • Abdelkader Zekhnini
    • 2
    Email author
  • Mohammed Taous
    • 3
  1. 1.Mathematics Department, Sciences FacultyMohammed First UniversityOujdaMorocco
  2. 2.Mathematics Department, Pluridisciplinary faculty of NadorMohammed First UniversityNadorMorocco
  3. 3.Mathematics Department, Sciences and Techniques FacultyMoulay Ismail UniversityErrachidiaMorocco

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