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The paper presents a classification of quadratic extension algebras, alsoknown as algebras of degree 2, as well as several characterizations ofquaternion algebras over a field (of characteristic not 2). The presentation isnot restricted to finite-dimensional algebras. The 'pure calculus' on aquaternion algebra is introduced; it generalizes the 'vector calculus' of theHamilton's quaternions, and is instrumental in establishing, in a clear andsimple way, the isomorphism between the group consisting of all automorphismsand all anti-automorphisms of a quaternion algebra and the orthogonal group ofthe norm form on the subspace of pure quaternions. The paper concludes with aglimpse of quaternion algebras over an integral domain (of characteristic not2), and of their facility in studying ternary quadratic forms over an integraldomain.

Quadratic Extension Algebras and Quaternion Algebras Over Fields (of Characteristic not 2)