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


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 (∗).


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 



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


  1. 1.
    Azizi, A.: Sur la capitulation des 2-classes d’idéaux de \(\mathbb {k}=\mathbb {Q}(\sqrt 2pq,i)\), où p≡−q≡1 (mod 4). Acta. Arith. 94, 383–399 (2000)MathSciNetGoogle Scholar
  2. 2.
    Azizi, A.: Unités de certains corps de nombres imaginaires et abéliens sur \(\mathbb {Q}\). Ann. Sci. Math. Québec 23(1), 15–21 (1999)MathSciNetGoogle Scholar
  3. 3.
    Azizi, A., Taous, M.: Détermination des corps \(\mathbf {k}=\mathbb {Q}(\sqrt {d}, \sqrt {-1})\) dont les 2-groupes de classes sont de type (2,4) ou (2,2,2). Rend. Istit. Mat. Univ. Trieste. 40, 93–116 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Azizi, A., Zekhnini, A., Taous, M.: On the generators of the 2-class group of the field \(\mathbb {Q}(\sqrt {d},i)\). IJPAM 81(5), 773–784 (2012)Google Scholar
  5. 5.
    Zekhnini, A., Azizi, A., Taous, M.: On the generators of the 2-class group of the field \(\mathbb {Q}(\sqrt {q_1q_2p,}i)\) Correction to Theorem 3 of [5]. IJPAM 103(1), 99–107 (2015)Google Scholar
  6. 6.
    Azizi, A., Zekhnini, A., Taous, M.: On the strongly ambiguous classes of \(\mathbb {k}/\mathbb {Q}(i)\) where \(\mathbb {k}=\mathbb {Q}(\sqrt {2p_1p_2,} i)\). Asian-Eur. J. Math. 7(1), 26 (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Azizi, A., Zekhnini, A., Taous, M.: On the strongly ambiguous classes of some biquadratic number fields. Math. Bohem. doi: 10.21136/MB.2016.0022-14(2016)
  8. 8.
    Azizi, A., Zekhnini, A., Taous, M.: Capitulation in the absolutely abelian extensions of some fields \(\mathbb {Q}(\sqrt {p_1p_2q,} \sqrt {-1})\). 1507.00295v1. Submitted
  9. 9.
    Azizi, A., Zekhnini, A., Taous, M.: Structure of Gal(𝕜 (2)2/𝕜) for some fields \(\mathbb {k}=\mathbb {Q}(\sqrt {2p_1p_2,} i)\) with C l 2(𝕜)≃(2,2,2). Abh. Math. Sem. Univ. Hamburg. 84(2), 203–231 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Azizi, A., Zekhnini, A., Taous, M., Mayer, D. C.: Principalization of 2-class groups of type (2,2,2) of biquadratic fields \(\mathbb {Q}(\sqrt {p_1p_2q,} i)\). Int. J. Number Theory 11(4), 1177–1215 (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    Azizi, A., Zekhnini, A., Taous, M.: Coclass of \(\text {Gal}(\mathbb {k}_{2}^{(2)}/\mathbb {k})\) for some fields \(\mathbb {k}=\mathbb {Q}\left (\sqrt p_1p_2q, i\right )\) with 2-class groups of type (2,2,2). J. Algebra Appl. 15(2), 26 (2016)CrossRefGoogle Scholar
  12. 12.
    Chevalley, C.: Sur la théorie du corps de classes dans les corps finis et les corps locaux. Thèses franaises de l’entre-deux-guerres 155, 365–476 (1934)zbMATHGoogle Scholar
  13. 13.
    Lemmermeyer, F.: The ambiguous class number formula revisited. J. Ramanujan Math. Soc. 28(4), 415–421 (2013)MathSciNetGoogle Scholar
  14. 14.
    Heider, F. P., Schmithals, B.: Zur kapitulation der idealklassen in unverzweigten primzyklischen erweiterungen. J. Reine Angew. Math. 366, 1–25 (1982)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Terada, F.: A principal ideal theorem in the genus fields. Tohoku Math. J. 23 (2), 697–718 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Furuya, H.: Principal ideal theorems in the genus field for absolutely abelian extensions. J. Number Theory 9, 4–15 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hasse, H.: Über die Klassenzahl abelscher Zahlkörper (1952)Google Scholar
  18. 18.
    Wada, H.: On the class number and the unit group of certain algebraic number fields. J. Fac. Sci. Univ. Tokyo Sect. I(13), 201–209 (1966)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sime, P. J.: On the ideal class group of real biquadratic fields. Trans. Am. Math. Soc. 347(12), 4855–4876 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Louboutin, S.: Hasse unit indices of dihedral octic CM-fields. Math. Nachr. 215, 107–113 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kubota, T.: Über den bizyklischen biquadratischen Zahlkörper. Nagoya Math. J. 10, 65–85 (1956)CrossRefzbMATHGoogle Scholar
  22. 22.
    McCall, T. M., Parry, C. J., Ranalli, R. R.: Imaginary bicyclic biquadratic fields with cyclic 2-class group. J. Number Theory 53, 88–99 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations