Let m be an integer 3 and let F be one of the fields or . Denote the Galois group Gal(F/) by Δ and let p be an odd prime such that p |Δ|, where |Δ| denotes the order of Δ. Let Δ denote the p-part of the ideal class group of F and (E/C)p denote the p-part of the group E of units of F modulo the subgroup C of cyclotomic units.;For F = assuming certain conditions for m and p, the equality |eρA = is obtained, where eρ is the idempotent corresponding to an irreducible higher dimensional odd character ρ of Δ into distinct from the Teichmüller character and is the highest power of p dividing the p -adic integer , which is defined in terms of generalized Bernoulli numbers.;The method applied is an extension of Rubin's early treatment of Kolyvagin's Euler systems.;For F = , the equality |eρΔ| = | eρ(E/C)p| is proven, where eρ is the idempotent corresponding to an irreducible higher dimensional character p of Δ into . Furthermore, it is shown that eρ (E/C)p is a principal [Δ]-module.
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