Automorphisms of local fields of period $p^M$ and nilpotent class $<p$

Abstract

Suppose $K$ is a finite extension of ${\mathbb{Q}}_p$ containing a $p^M$-th primitive root of unity. For $1⩽s<p$ 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 ${\Gamma }_K^{(v)}$ 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.

arXiv comment27 pages; revised version