Advertisement

manuscripta mathematica

, Volume 72, Issue 1, pp 307–324 | Cite as

Lois de reciprocite primitives

  • Nguyen Quang Do Thong
Article

Abstract

For an algebraic number fieldK containing roots of unity of orderp m (p a prime), we study a new kind of reciprocity law, called “primitive”, which allows us, when it exists, to express wild Hilbert symbols of orderp m in terms of a (uniformly) finite set of tame symbols. A sufficient condition for existence is the nullity of thep-group of classes ofp-ideals ofK. Applications are given to the description of the Galois group of the maximal prop-extension ofK which is unramified outside some finite set of primes containingp.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. [1]
    J.-W. Cassels &A. Fröhlich (ed.):Algebraic Number Theory, Academic Press, London (1967)zbMATHGoogle Scholar
  2. [2]
    G. Gras &J.-F. Jaulent:Sur les corps de nombres réguliers, Math. Z. 202, 343–365 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J.-F. Jaulent: L’arithmétique des ℓ-extensions Thèse d’Etat, Besançon (1986)Google Scholar
  4. [4]
    H. Koch:Galoissche Theorie der p-Erweiterungen, Deutsch. Verlag der Wissen., Berlin (1970)zbMATHGoogle Scholar
  5. [5]
    A. Movahhedi &T. Nguyen Quang Do:Sur l’arithmétique des corps de nombres p-rationnels, Sém. Théorie des Nombres Prris 1987–88, Birkhäuser vol.81, 155–200 (1990)MathSciNetGoogle Scholar
  6. [6]
    T. Nguyen Quang Do: Sur la ℤp-torsion de certains modules galoisiens,Ann. Inst. Fourier 36, 2, 27–46 (1986)zbMATHMathSciNetGoogle Scholar
  7. [7]
    J. Neukirch:Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Arch. der Math. 22, 337–357 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J.-P. Serre:Structure de certains pro-p-groupes, Séminaire Bourbaki n o 252, 11 p. (1963).Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Nguyen Quang Do Thong
    • 1
  1. 1.URA 741 CNRS Laboratoire de MathématiquesUniversité de Franche-ComtéBESANCON Cédex

Personalised recommendations