Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Sur une question de capitulation


Author: Abdelmalek Azizi
Journal: Proc. Amer. Math. Soc. 130 (2002), 2197-2202
MSC (2000): Primary 11R37
DOI: https://doi.org/10.1090/S0002-9939-02-06424-9
Published electronically: January 31, 2002
MathSciNet review: 1897477
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $p$ and $q$ be prime numbers such that $p \equiv 1 \bmod 8,\,\, q \equiv -1 \bmod 4$and $(\frac{\textstyle p}{\textstyle q}) = - 1$. Let $d = pq$, $\mathbf{k} = \mathbf{Q}(\sqrt{d},i)$, and let $\mathbf{ k}^{(1)}_{2}$ be the 2-Hilbert class field of $\mathbf{k}$, $\mathbf{k}^{(2)}_{2}$ the 2-Hilbert class field of $\mathbf{k}^{(1)}_{2}$ and $G_{2}$ the Galois group of $\mathbf{k}^{(2)}_{2}/\mathbf{k}$. The 2-part $C_{\mathbf{k},2}$ of the class group of $\mathbf{ k}$ is of type $(2,2)$, so $\mathbf{k}_{2}^{(1)}$contains three extensions $\mathbf{K}_{i}/\mathbf{k},\,\,i = 1,\,2,\,3$. Our goal is to study the problem of capitulation of the 2-classes of $\mathbf{k}$ in $\mathbf{K}_{i},\,\,i = 1,\,2,\,3$, and to determine the structure of $G_{2}$.

RSESUM´E. Soient $p$ et $q$ deux nombres premiers tels que $p \equiv 1 \bmod 8,\,\, q \equiv -1\bmod 4$et $(\frac{\textstyle p}{\textstyle q}) = - 1$, $d = pq$, $i = \sqrt{-1}$, $\mathbf{k} = \mathbf{Q}(\sqrt{d},i)$, $\mathbf{k}^{(1)}_{2}$ le 2-corps de classes de Hilbert de $\mathbf{k}$, $\mathbf{k}^{(2)}_{2}$le 2-corps de classes de Hilbert de $\mathbf{k}^{(1)}_{2}$et $G_{2}$ le groupe de Galois de $\mathbf{k}^{(2)}_{2}/\mathbf{k}$. La 2-partie $C_{\mathbf{k}, 2}$du groupe de classes de $\mathbf{k}$est de type $(2,2)$, par suite $\mathbf{k}^{(1)}_{2}$contient trois extensions $\mathbf{K}_{i}/\mathbf{k},\,\,i = 1,\,2,\,3$. On s'intéresse au problème de capitulation des 2-classes de $\mathbf{k}$ dans $\mathbf{K}_{i},\,\,i = 1,\,2,\,3$, et à déterminer la structure de $G_{2}$.


References [Enhancements On Off] (What's this?)

  • 1.
    A. Azizi, Sur la capitulation des 2-classes d'idéaux de $\mathbf{ Q}(\sqrt d,i)$.
    C. R. Acad. Sci. Paris, t. 325, Série I, p. 127-130, (1997). MR 98d:11131
  • 2.
    A. Azizi, Sur le $2$-groupe de classes d'idéaux de $\mathbf{ Q}(\sqrt{d}, i)$.
    Rendiconti del circolo matematico di Palermo (2) 48 (1999), 71-92. MR 2000d:11131
  • 3.
    A. Azizi, Unités de certains corps de nombres imaginaires et abéliens sur $\mathbf{ Q}$.
    Annales des Sciences Mathématiques du Québec 23( 1999), no. 1, 15-21. MR 2000k:11120
  • 4.
    A. Azizi, Capitulation of the 2-ideal classes of $\mathbf{ Q}(\sqrt{p_1}, \sqrt{-p_2})$.
    Lecture notes in pure and applied mathematics, vol. 208, p. 13-19, 1999. MR 2000h:11118
  • 5.
    A. Azizi, Capitulation des 2-classes d'idéaux de $\mathbf{ Q}(\sqrt{2pq}, i)$.
    Acta Arithmetica 94 (2000), p 383 - 399. MR 2001k:11221
  • 6.
    P. Barruccand et H. Cohn, Note on primes of type $ x^{2} + 32y^{2}$, class number, and residuacity.
    J. reine angew. Math. 238 (1969), 67-70. MR 40:2641
  • 7.
    S. M. Chang and R. Foote, Capitulation in Class Field Extensions of Type (p,p).
    Can. J. Math. vol 32, No.5, (1980), 1229-1243. MR 82i:12013
  • 8.
    H. Cohn, The Explicit Hilbert 2-Cyclic Class Fields of $\mathbf{ Q}(\sqrt{-p})$.
    J. reine angew. Math. 321 (1981), 64-77. MR 82e:12011
  • 9.
    F. P. Heider und B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen.
    J. reine angew. Math. 336 (1982), 1-25. MR 84g:12002
  • 10.
    H. Kisilevsky, Number Fields with Class Number Congruent to 4 mod 8 and Hilbert's Theorem 94.
    J. Number Theory 8, (1976), 271-279. MR 54:5188
  • 11.
    T. Kubota, Über den bizyklischen biquadratischen Zahlkörper.
    Nagoya Math. J, 10 (1956), 65-85. MR 18:643e
  • 12.
    K. Miyake, Algebraic Investigations of Hilbert's Theorem 94,the Principal Ideal Theorem and Capitulation Problem.
    Expos. Math. 7 (1989), 289-346. MR 90k:11144
  • 13.
    H. Suzuki, A Generalisation of Hilbert's Theorem 94.
    Nagoya Math. J., vol 121 (1991). MR 92h:11098
  • 14.
    F. Terada, A Principal Ideal Theorem in the Genus Fields.
    Tôhoku Math. J. Second Series, vol. 23, 1971, pp. 697-718. MR 46:5285
  • 15.
    H. Wada, On the Class Number and the Unit Group of Certain Algebraic Number Fields.
    Tokyo U., Fac. of Sc. J., Serie I, 13 (1966), 201-209. MR 35:5414

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R37

Retrieve articles in all journals with MSC (2000): 11R37


Additional Information

Abdelmalek Azizi
Affiliation: Département de Mathématiques, Faculté des Sciences, Université Mohammed 1, Oujda, Maroc
Email: azizi@sciences.univ-oujda.ac.ma

DOI: https://doi.org/10.1090/S0002-9939-02-06424-9
Keywords: Groupe des unit\'es, syst\`eme fondamental d'unit\'es, capitulation, corps de classes de Hilbert
Received by editor(s): February 23, 2001
Published electronically: January 31, 2002
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society