An algorithm for the principal ideal problem in indefinite quaternion algebras

TL;DR

This paper introduces a heuristically subexponential algorithm for finding generators of principal ideals in indefinite quaternion algebras, addressing a key computational challenge in number theory with broad applications.

Contribution

The paper presents a novel heuristic subexponential algorithm for the principal ideal problem in indefinite quaternion algebras, improving computational methods in algebraic number theory.

Findings

Algorithm is heuristically subexponential in complexity

Effectively reduces the problem to the underlying number field

Provides practical approach for generator computation in quaternion algebras

Abstract

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.