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

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Quadratic and Higher Degree Forms

Part of the book series: Developments in Mathematics ((DEVM,volume 31))

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 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.

Research partially supported by NSA Grant H98230-09-1-0037 and NSF Grant DMS-0901971.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT, 05401, USA

    John Voight

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  1. John Voight
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Correspondence to John Voight .

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Editors and Affiliations

  1. Department of Mathematics, University of Florida Department of Mathematics, Gainesville, Florida, USA

    Krishnaswami Alladi

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Manjul Bhargava

  3. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    David Savitt

  4. Department of Mathematics, University of Arizona, Tuscon, Arizona, USA

    Pham Huu Tiep

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Voight, J. (2013). Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P. (eds) Quadratic and Higher Degree Forms. Developments in Mathematics, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7488-3_10

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