  Suppose $K$ is a finite extension of $\mathbb{Q}_p$ containing a $p^M$-th
primitive root of unity. For $1\leqslant s<p$ denote by $K[s,M]$ the maximal
$p$-extension of $K$ with the Galois group of period $p^M$ and nilpotent class
$s$. We apply the nilpotent Artin-Schreier theory together with the theory of
the field-of-norms functor to give an explicit description of the Galois groups
${Gal}(K[s,M]/K)$. As application we prove that the ramification subgroup
$\Gamma ^{(v)}_K$ of the absolute Galois group of $K$ acts trivially on
$K[s,M]$ if and only if $v>e_K(M+s/(p-1))-(1-\delta _{1s})/p$, where $e_K$ is
the ramification index of $K$ and $\delta _{1s}$ is the Kronecker symbol.
