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

Abstract: The paper presents a classification of quadratic extension algebras, also known as algebras of degree 2, as well as several characterizations of quaternion algebras over a field (of characteristic not 2). The presentation is not restricted to finite-dimensional algebras. The 'pure calculus' on a quaternion algebra is introduced; it generalizes the 'vector calculus' of the Hamilton's quaternions, and is instrumental in establishing, in a clear and simple way, the isomorphism between the group consisting of all automorphisms and all anti-automorphisms of a quaternion algebra and the orthogonal group of the norm form on the subspace of pure quaternions. The paper concludes with a glimpse of quaternion algebras over an integral domain (of characteristic not 2), and of their facility in studying ternary quadratic forms over an integral domain.