Classification of Real Division Algebras
DOI:
https://doi.org/10.15381/pesquimat.v26i2.25686Keywords:
algebra of division, conjugation, quaternion, octonionAbstract
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.
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Copyright (c) 2023 Wilber Carrillo Flores, Alberto Mariano Rivero Zapata
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