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. Since the discovery of the division ring of quaternions over the real numbers by Hamilton, and continuing with work of Albert and many others, a deep link has been forged between quadratic forms in three and four variables over a field F and quaternion algebras over F. Starting with a quaternion algebra over F, a central simple F-algebra of dimension 4, one obtains a quadratic form via the reduced norm (restricted to the trace zero subspace); the split quaternion algebra over F, the 2 × 2-matrix ring M2(F), corresponds to an isotropic quadratic form, one that represents zero nontrivially. (Conversely, one recovers the quaternio
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