Galois theoretic local-global relations in nilpotent extensions of algebraic number fields

  • Katsuya Miyake
Part of the Progress in Mathematics book series (PM, volume 102)


The Galois group of the maximal abelian extension of an algebraic number field of finite degree is excellently described by class field theory with ideles of the ground field, which is due to C. Chevalley. We see by it how the local arithmetic phenomena are tied up as a glogal whole by the relations determined by the global numbers. The purpose of this note is to give an “analogous” description of the Galois group of the maximal nilpotent extensions of the ground field which may be regarded as a “Galois theoretic lifting” of the abelian case.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ch]
    C. Chevalley. — Deux théorèmes d’arithmétique, J. Math. Soc. Japan 3, (1951), 36 - 44.MathSciNetzbMATHGoogle Scholar
  2. [F]
    A. Frolich. — Central extensions, Galois groups and ideal class groups of number fields, Contemporary Math. 24, A.M.S., 1983.Google Scholar
  3. [K]
    H. Koch. — Galoissche Theorie der p-Erweiterungen, Deutsch-Verlag Wissen., Berlin, 1970.CrossRefzbMATHGoogle Scholar
  4. [M-Ng]
    A. Movahhedi et T. Nguyen Quang Do. — Sur l’arithmétique des corps de nombres p-rationnels, in: C. Goldstein (ed.), Séminaire de Théorie des Nombres de Paris, 1987 — 88, Birkhäuser, 1990, 155 - 200.Google Scholar
  5. [S]
    J.-P. Serre. — Modular forms of weight one and Galois representations, in: A. Fröhlich (ed.), Algebraic number fields, Acad Press, 1977, 193 - 268.Google Scholar
  6. [Sh]
    G. Shimura. — On canonical models of arithmetic quotients of bounded symmetric domains II, Ann. of Math. 92, (1970), 528 - 549.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Katsuya Miyake
    • 1
  1. 1.Department of MathematicsCollege of General Education Nagoya UniversityChikusa-ku, NagoyaJapan

Personalised recommendations