Finite local principal ideal rings

TL;DR

This paper explores the structure of finite local principal ideal rings, showing they are homomorphic images of discrete valuation rings and introducing a group action on Eisenstein polynomials to analyze their properties.

Contribution

It establishes the connection between finite local principal ideal rings and discrete valuation rings, and introduces a novel group action on Eisenstein polynomials for their study.

Findings

Finite local principal ideal rings are homomorphic images of discrete valuation rings.

A non-commutative group acts on Eisenstein polynomials via resultants.

The invariants of these rings are characterized by five parameters.

Abstract

Every finite local principal ideal ring is the homomorphic image of a discrete valuation ring of a number field, and is determined by five invariants. We present an action of a group, non-commutative in general, on the set of Eisenstein polynomials, of degree matching the ramification index of the ring, over the coefficient ring. The action is defined by taking resultants.