Normes cyclotomiques na{\"i}ves et unit{\'e}s logarithmiques

TL;DR

This paper investigates the structure of norm groups in cyclotomic Z{ extellipsis}-extensions of number fields, comparing them with logarithmic units, and provides a simple proof of the Gross-Kuz'min conjecture in certain cases.

Contribution

It introduces a method to compute the Z-rank of norm subgroups and offers an easy proof of the Gross-Kuz'min conjecture for { extellipsis}-undecomposed abelian extensions.

Findings

Computed the Z-rank of norm subgroups in cyclotomic extensions.

Compared { extellipsis}-adification with logarithmic units.

Provided a simple proof of the Gross-Kuz'min conjecture for specific extensions.

Abstract

We compute the Z-rank of the subgroup of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic Z{\ell}-extension of K. Thus we compare its {\ell}-adification with the group of logarithmic units of K. By the way we point out an easy proof of the Gross-Kuz'min conjecture for {\ell}-undecomposed extensions of abelian fields.