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

G{\'e}n{\'e}ralisation d'un th{\'e}or{\`e}me de Greenberg