  We formulate a general conjecture on the characteristic polynomials of
S-decomposed T-ramified Iwasawa modules over the cyclotomic Z {\ell}-extension
of a number field. We show that this conjecture is equivalent to the
conjunctions of the classical conjectures of Leopoldt and of Gross-Kuz'min. We
so extend a result of semi-simplicity of Greenberg and, by the way, an
isomorphism of Kuz'min.
