Extended gcd of quadratic integers

TL;DR

This paper presents a method for computing the extended gcd of two quadratic integers using properties of principal ideal rings and binary quadratic forms, applicable regardless of Euclidean properties.

Contribution

It introduces a novel approach leveraging principal ideal rings and quadratic form reduction for extended gcd computation in quadratic integer rings.

Findings

Effective extended gcd computation in principal ideal quadratic integer rings

Method applicable to both Euclidean and non-Euclidean rings

Potential for broader applications in algebraic number theory

Abstract

Computation of the extended gcd of two quadratic integers. The ring of integers considered is principal but could be euclidean or not euclidean ring. This method rely on principal ideal ring and reduction of binary quadratic forms.