Kolyvagin’s System of Gauss Sums
Recently in  Kolyvagin introduced a remarkable inductive procedure which improves upon Stickelberger’s theorem and results of Thaine  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 , 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.
KeywordsDirichlet Character Main Conjecture Ideal Class Group Cyclotomic Field Abelian Group Structure
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