Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms

TL;DR

This paper explores algorithms for recognizing quaternion algebras and quadratic forms, focusing on embedding rank 4 algebras into matrix rings and computing related invariants, with applications in computational algebra.

Contribution

It introduces algorithms for identifying quaternion algebras among rank 4 algebras and computing their structural invariants, advancing computational methods in algebra.

Findings

Algorithms for embedding rank 4 algebras into M_2(R)

Methods for computing the Hilbert symbol

Procedures for determining maximal orders

Abstract

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2x2-matrix ring M_2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.