A converse to the Hasse-Arf theorem

TL;DR

The paper proves a converse to the Hasse-Arf theorem, showing that nonabelian Galois groups can have extensions with nonintegral upper ramification breaks, unlike abelian groups.

Contribution

It establishes that nonabelian Galois groups can exhibit nonintegral ramification breaks, providing a converse perspective to the classical Hasse-Arf theorem.

Findings

Nonabelian Galois groups can have nonintegral upper ramification breaks.

Constructs explicit examples of such extensions.

Extends understanding of ramification in local field extensions.

Abstract

Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$ be a nonabelian group which is isomorphic to the Galois group of some totally ramified extension $E/F$ of local fields with residue characteristic $p>2$. Then there is a totally ramified extension of local fields $L/K$ with residue characteristic $p$ such that Gal$(L/K)\cong G$ and $L/K$ has at least one nonintegral upper ramification break.