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

TL;DR

This paper explicitly describes the Galois groups of certain $p$-extensions of local fields using nilpotent Artin-Schreier theory and field-of-norms, and characterizes ramification subgroup actions.

Contribution

It provides an explicit description of Galois groups of maximal $p$-extensions with specified period and nilpotent class, extending understanding of local field automorphisms.

Findings

Explicit Galois group descriptions for $K[s,M]$ extensions.

Characterization of ramification subgroup triviality conditions.

Application of nilpotent Artin-Schreier and field-of-norms theories.

Abstract

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.