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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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