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

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

2010-04-07

arxiv.org/abs/1004.0994v1

Mathematics

Number Theory

35 pages

Scientific paper

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.

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