Reduced ideals from the reduction algorithm

TL;DR

This paper investigates the limitations of the reduction algorithm in computing reduced ideals in number fields, identifying conditions under which certain ideals can or cannot be obtained, especially in real quadratic fields.

Contribution

It provides a necessary and sufficient condition for reduced ideals of real quadratic fields to be generated by the reduction algorithm.

Findings

Some reduced ideals cannot be obtained from the reduction algorithm.

Ideals with inverses of larger norms among reduced ideals are not generated by the algorithm.

A characterization for when reduced ideals are obtainable from the algorithm in real quadratic fields.

Abstract

The reduction algorithm is used to compute reduced ideals of a number field. However, there are reduced ideals that can never be obtained from this algorithm. In this paper, we will show that these ideals have inverses of larger norms among reduced ones. Especially, we represent a sufficient and necessary condition for reduced ideals of real quadratic fields to be obtained from the reduction algorithm.