Extensions of local fields given by 3-term Eisenstein polynomials

TL;DR

This paper classifies certain totally ramified local field extensions with two indices of inseparability, showing they can be described by Eisenstein polynomials with at most three terms, extending previous work for degree p extensions.

Contribution

It provides an explicit classification of extensions with two indices of inseparability using 3-term Eisenstein polynomials, generalizing Amano's results for degree p extensions.

Findings

Existence of a uniformizer with a 3-term minimal polynomial

Classification of extensions with two indices of inseparability

Extension of Amano's work to higher degrees

Abstract

Let $K$ be a local field with residue characteristic $p$ and let $L/K$ be a totally ramified extension of degree $p^k$. In this paper we show that if $L/K$ has only two distinct indices of inseparability then there exists a uniformizer $\pi_L$ for $L$ whose minimum polynomial over $K$ has at most three terms. This leads to an explicit classification of extensions with two indices of inseparability. Our classification extends work of Amano, who considered the case $k=1$.