Computing in quotients of rings of integers

TL;DR

This paper introduces algorithms to transform quotients of rings of integers into Euclidean rings and develops a modular normal form algorithm for modules, significantly improving computational efficiency over classical methods.

Contribution

It presents novel polynomial algorithms for fundamental operations in quotients of rings of integers and a new modular normal form algorithm for modules over these rings.

Findings

Algorithms effectively convert quotients into Euclidean rings

New modular normal form algorithm outperforms classical methods

Enhanced computational tools for modules over rings of integers

Abstract

We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.