RÉSUMÉ

Let k be a number field, K the compositum of all Z_p-extensions of k, p a prime number, and A the inverse limit of the p-part of the ideal class group over all finite subextensions of K/k. Greenberg has conjectured that A is pseudo-null (that is, of codimension at least 2) as a module over the Iwasawa algebra of K/k. We prove this conjecture in the case that k is the field of p-th roots of unity, p exactly divides the class number of k, and a certain mild additional hypothesis on units is satisfied.

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