Artin-Schreier-Witt extensions and ramification breaks

Abstract

Let $K=k((t))$ be a local field of characteristic $p>0$, with perfect residue field $k$. Let $\vec{a}=(a_0,a_1,\dots,a_{n-1})\in W_n(K)$ be a Witt vector of length $n$. Artin-Schreier-Witt theory associates to $\vec{a}$ a cyclic extension $L/K$ of degree $p^i$ for some $i\le n$. Assume that the vector $\vec{a}$ is ``reduced'', and that $v_K(a_0)<0$; then $L/K$ is a totally ramified extension of degree $p^n$. In the case where $k$ is finite, Kanesaka-Sekiguchi and Thomas used class field theory to explicitly compute the upper ramification breaks of $L/K$ in terms of the valuations of the components of $\vec{a}$. In this note we use a direct method to show that these formulas remain valid when $k$ is an arbitrary perfect field.