Division algorithms for norm-Euclidean imaginary quadratic fields

TL;DR

This paper provides division algorithms for all known norm-Euclidean imaginary quadratic fields, enabling efficient computation of remainders within these fields.

Contribution

It introduces explicit division algorithms for each known norm-Euclidean imaginary quadratic field, enhancing computational methods in algebraic number theory.

Findings

Finite list of norm-Euclidean imaginary quadratic fields is known.

Provides division algorithms for each known case.

Algorithms find remainders within the Euclidean minimum.

Abstract

The list of norm-Euclidean imaginary quadratic fields is known and finite. For each known case, we give a division algorithm that finds a remainder at distance less than the Euclidean minimum of the field.