On a Special Ideal Contained in the Stickelberger Ideal

Abstract

In this paper an ideal B of the group ring R=Z[G ] generated by a special Kummer element is studied (G means a cyclic group of order l&1 and l and odd prime). The group index [R*: B] is expressed by means of the relative class number of the l th cyclotomic field (R* a subring of R). The ideal B is investigated also mod l. The connection with the (modified) Demjanenko and the Benneton system of congruences is mentioned.

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“…This result is considered as a natural generalization of the class number formula for the p-th cyclotomic field, given in our previous article [3]. In view of the fact that our previous formula plays a certain role in the study of this field (see [7], [9], [10]), we believe the present formula would give some insight in the investigation of the arithmetic of the pq-th cyclotomic field.…”

Determinantal Formula for the Special Values of the Dedekind Zeta Function of the Cyclotomic Field

“…Silverman's matrix) is in fact, intimately related with the first (resp. second) Stickelberger's element introduced in [1] (see also [6])…”

Stickelberger Subideals for a Prime Modulus Related to Kummer-Type Congruences

“…The system (K(1 )) for the special case where 1= [2] was first observed by Benneton [4] and it was recently investigated by Skula [8] standing on the point of view of the Stickelberger ideal in a group ring. More general cases with *1=1 and 2 were treated in the papers [2,3].…”