A family of Eisenstein polynomials generating totally ramified extensions, identification of extensions and construction of class fields

TL;DR

This paper introduces a normal form for Eisenstein polynomials over local fields, classifies all totally ramified extensions they generate, and provides algorithms for identifying and constructing class fields from these polynomials.

Contribution

It defines a normal form for Eisenstein polynomials, classifies all totally ramified extensions they generate, and develops algorithms for extension identification and class field construction.

Findings

Normal form for Eisenstein polynomials depending on uniformizer and residue representatives

Exhaustive classification of totally ramified extensions generated by these polynomials

Algorithms for extension recovery and class field construction

Abstract

Let $K$ be a local field with finite residue field, we define a normal form for Eisenstein polynomials depending on the choice of a uniformizer $\pi_K$ and of residue representatives. The isomorphism classes of extensions generated by the polynomials in the family exhaust all totally ramified extensions, and the multiplicity with which each isomorphism class $L/K$ appears is always smaller than the number of conjugates of $L$ over $K$. An algorithm to recover the set of all special polynomials generating the extension determined by a general Eisenstein polynomial is described. We also give a criterion to quickly establish that a polynomial generates a different extension from that generated by a set of special polynomials, such criterion does not only depend on the usual distance on the set of Eisenstein polynomials considered by Krasner and others. We conclude with an algorithm for the construction of the unique special equation determining a totally ramified class field in general degree, given a suitable representation of a group of norms.