Kolyvagin’s System of Gauss Sums

  • Karl Rubin
Part of the Progress in Mathematics book series (PM, volume 89)


Recently in [1] Kolyvagin introduced a remarkable inductive procedure which improves upon Stickelberger’s theorem and results of Thaine [6] on ideal class groups of cyclotomic fields. For every Dirichlet character x modulo a prime p Kolyvagin was able to determine the order of the x-component of the p-part of the ideal class group of Q(μ p ). These orders were already known from the work of Mazur and Wiles [3], but Kolyvagin’s proof is very much simpler. Kolyvagin’s method also determines the abelian group structure of these ideal class groups in terms of Stickelberger ideals.


Dirichlet Character Main Conjecture Ideal Class Group Cyclotomic Field Abelian Group Structure 
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  1. [1]
    V. A. Kolyvagin: Euler systems. To appear.Google Scholar
  2. [2]
    S. Lang: Cyciotomic Fields. Grad. Texts in Math. 59, Springer-Verlag: New York (1978) (also 2nd edition, Grad. Texts in Math. 121, Springer-Verlag: New York (1990)).CrossRefGoogle Scholar
  3. [3]
    B. Mazur, A. Wiles: Class fields of abelian extensions of Q. Invent. Math. 76 (1984) 179–330.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    K. Rubin: The Main Conjecture. Appendix to: Cyclotomic Fields I and II, combined 2nd edition. Grad. Texts in Math. 121, Springer-Verlag: New York (1990) 397–419.Google Scholar
  5. [5]
    R. Schoof: The structure of the minus class groups of abelian number fields. University Utrecht preprint #588, October 1989.Google Scholar
  6. [6]
    F. Thaine: On the ideal class groups of real abelian number fields. Annals of Math. 128 (1988) 1–18.MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 1991

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  • Karl Rubin

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