For an abelian number field K containing a primitive pth root of
unity (p an odd prime) and satisfying certain technical conditions, we
parametrize the [Z.sub.p] [G(K/Q)]-annihilators of the "minus"
part [A.sup.-.sub.K] of the p-class group by means of modules of Jacobi
sums. Using a reflection theorem and Bloch-Kato's reciprocity law,
we then determine the Fitting ideal of the "plus" part
[A.sup.+.sub.K] in terms of "twisted" Gauss sums.