Classification of Real Division Algebras

Abstract

This article aims to offer a unifying approach to the basic theory of division algebras by presenting the research of the German-American mathematician Max August Zorn, who classified alternative division algebras. In section 1 the basic theory of real division algebras is developed. Section 2 presents the Cayley-Dickson Process, which consists of constructing an extension algebra from an algebra provided with a conjugation, similar to the construction of complex numbers from real numbers. In Section 3 presents the classical division algebras R (real), C (complex), H (quaternions) and O (octonions) and mentions some of their applications. In section 4 the main theorem is presented, which establishes that the only (except isomorphism) alternative division algebras are: R, C, H and O (Zorn’s theorem). The classification theorems of associative division algebras (Frobenius) and normed division algebras (Hurwitz) are obtained as corollaries of Zorn’s theorem. Finally in section 5 applications of division algebras to Geometry, Number Theory, Classical Physics, Modern Physics, Quantum Mechanics and Cryptography are mentioned.