Semi-local modulo unities of Gauss sums

For an odd prime number p and an abelian number field k, let k(infinity),k be the cyclotomic Z(p)-extension. Let X-infinity be the projective limit of the p-parts of the ideal class groups of each intermediate field of k(infinity)/k. It is conjectured (Greenberg's Conjecture) that X-infinity is finite when k is totally real. In this paper we give an interpretation of the characteristic polynomial of X-infinity in terms of certain Gauss sums. We also give analogous results at finite level. Our results generalize those obtained by Ichimura (J. Number Theory 68 (1998) 36) and Hachimori (Manuscripta Math. 95 (1998) 377) in the semi-simple case. (c) 2005 Elsevier Inc. All rights reserved.