An identification for Eisenstein polynomials over a p-adic field

TL;DR

This paper provides criteria to determine when two Eisenstein polynomials over a p-adic field define the same extension, and fully characterizes Eisenstein polynomials of degree p.

Contribution

It introduces a criterion for identifying when Eisenstein polynomials over p-adic fields define identical extensions and completely classifies degree p Eisenstein polynomials.

Findings

Criteria for polynomial extension equivalence

Complete classification of degree p Eisenstein polynomials

Translation of original results into English

Abstract

In this note, we give a criteria whether given two Eisenstein polynomials over a padic field define the same extension (Proposition 1.6). In particular, we completely identify Eisenstein polynomials of degree p (Theorem 1.16). This note is an English translation of a part of the original paper [12].