On enumerating extensions of p-adic fields with given invariants

TL;DR

This paper revisits the theory of ramification polygons of Eisenstein polynomials over p-adic fields, providing algorithms to enumerate all extensions with specified invariants and to generate all ramification polygons of a given degree.

Contribution

It offers a comprehensive re-exposition of existing theory and introduces algorithms for enumerating p-adic field extensions based on ramification invariants.

Findings

Algorithms to produce all extensions for a given ramification polygon

Method to generate all ramification polygons of a specified degree

Complete enumeration of totally ramified extensions of a given degree

Abstract

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given ramification polygon. We supplement this with an algorithm to produce all ramification polygons of a given degree, and hence we can produce all totally ramified extensions of a given degree.