Kolyvagin’s System of Gauss Sums

Abstract

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.KeywordsDirichlet CharacterMain ConjectureIdeal Class GroupCyclotomic FieldAbelian Group StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.