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.
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