Algorithm for computing the factor ring of an ideal in Dedekind domain with finite rank

TL;DR

This paper presents a deterministic polynomial-time algorithm for computing the factor ring of an ideal in a Dedekind domain with finite rank, enabling efficient prime ideal testing.

Contribution

It introduces a novel polynomial-time algorithm using basis representation, Hermite and Smith normal forms for Dedekind domains with finite rank.

Findings

Algorithm runs in deterministic polynomial time

Enables testing whether an ideal is prime or prime power

Uses basis representation with Hermite and Smith normal forms

Abstract

We give an algorithm for computing the factor ring of a given ideal in a Dedekind domain with finite rank, which runs in deterministic and polynomial-time. We provide two applications of the algorithm: judging whether a given ideal is prime or prime power. The main algorithm is based on basis representation of finite rings which is computed via Hermite and Smith normal forms.