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On formulae for the class number of real Abelian fields

For a given real Abelian field and a given prime natural number we obtain an index formula for the order of the group , where is the -component of the class group of denotes the -component of corresponding to a -irreducible character of the Galois group that is trivial on the Sylow -subgroup of . This result generalizes a conjecture of Gras. The proofs rely on the “main conjecture” of Iwasawa theory.