Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms

  • John VoightEmail author
Part of the Developments in Mathematics book series (DEVM, volume 31)


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 2 ×2-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.

Key words and Phrases

Quadratic forms Quaternion algebras Maximal orders Algorithms Matrix ring Number theory 

Mathematics Subject Classification (2010):

Primary 11R52 Secondary 11E12 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

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