Artin-Schreier-Witt extensions and ramification breaks
G. Griffith Elder, Kevin Keating
TL;DR
The paper addresses the ramification structure of Artin–Schreier–Witt extensions K_n/K over a local field K of characteristic p with perfect residue field, constructing these extensions from a Witt vector a∈W_n(K) by solving F(a)=a. It develops a direct, self-contained method to compute the upper ramification breaks in terms of valuations m_i=−v_K(a_i), extending prior finite-residue-field results to arbitrary perfect residue fields by reducing to strongly reduced Witt vectors and applying induction on n. The main contribution is the explicit formula u_i = max{ p^{i−1} m_0, ..., p m_{i−2}, m_{i−1} } for i=1,…,n, with clear compatibility with subextensions and quotients. This provides a concrete description of ramification in Artin–Schreier–Witt towers and has implications for explicit class field theory in characteristic p.
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.