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

2010/04/07 by John Voight, Voight, John · 2 citations
Computer Science · Engineering · Mathematics · #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1004.0994

openalex publication_date 2010/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 M₂(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.

Cited by

• Selmer ranks under quadratic twists satisfying the Heegner hypothesis
• Computing Hilbert modular forms as orthogonal modular forms

Related

• Characterizing quaternion rings over an arbitrary base
• Matrix Theory over the Complex Quaternion Algebra
• Ideal Class Group Algorithms in the Ring of Integral Quaternions
• Algebraic Groups Constructed from Orders of Quaternion Algebras
• Minimal Polynomials of Some Matrices Via Quaternions