On cyclotomic norms and the conjectures of Leopoldt and Gross-Kuz'min

We use $\mathrm{\ell }$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI.