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Sur la capitulation dans une \({\mathbb{Z}}_{\ell}\)-extension. (French) Zbl 0564.12011
Let \(K_{\infty}=\cup_{n\in {\mathbb{N}}}K_ n\) be a \({\mathbb{Z}}_{\ell}\)- extension of a number field K, and \(C\ell_{\infty}=\lim_{\to}C\ell_ n\) the \(\ell\)-class group of \(K_{\infty}\). We study the subgroup \(Cap_ n\) of ideal classes in \(C\ell_ n\) which become trivial in \(C\ell_{\infty}\). For n large enough, assuming that the Iwasawa invariant \(\mu\) vanishes, we prove that the groups \(Cap_ n\) are isomorphic to each other, and direct summands of \(C\ell_ n\).

MSC:
11R18 Cyclotomic extensions
11R23 Iwasawa theory
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