Hasse-Arf property and abelian extensions for local fields with imperfect residue fields

Abstract

For a finite totally ramified extension $L$ of a complete discrete valuation field $K$ with the perfect residue field of characteristic $p>0$, it is known that $L/K$ is an abelian extension if the upper ramification breaks are integers and if the wild inertia group is abelian. We prove a similar result without the assumption that the residue field is perfect. As an application, we prove a converse to the Hasse-Arf theorem for a complete discrete valuation field with the imperfect residue field. More precisely, for a complete discrete valuation field $K$ with the residue field $\overline{K}$ of residue characteristic $p>2$ and a finite non-abelian Galois extension $L/K$ such that the Galois group of $L/K$ is equal to the inertia group $I$ of $L/K$, we construct a complete discrete valuation field $K'$ with the residue field $\overline{K}$ and a finite Galois extension $L'/K'$ which has at least one non-integral upper ramification break and whose Galois group and inertia group are isomorphic to $I$.