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Naive cyclotomic norms and logarithmic units. (Normes cyclotomiques naïves et unités logarithmiques.) (French. English summary) Zbl 1430.11147

Summary: We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field \(K\) that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of \(K\). Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\mathcal E}_K\). By the way we point out an easy proof of the Gross-Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.

MSC:

11R23 Iwasawa theory
11R27 Units and factorization

Software:

PARI/GP
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Full Text: DOI

References:

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