Groups of automorphisms of local fields of period p^M and nilpotent class < p.

Abstract

Suppose K is a finite field extension of Qp containing a pM-th primitive root of unity. For 1 6 s < p denote by K[s, M] the maximal p-extension of K with the Galois group of period pM 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 of K[s, M]/K. As application we prove that the ramification subgroup of the absolute Galois group of K with the upper index v acts trivially on K[s, M] iff v > eK(M + s/(p − 1)) − (1 − δ1s)/p, where eK is the ramification index of K and δ1s is the Kronecker symbol.