Classification of Real Division Algebras
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 present...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | Perú |
| Institución: | Universidad Nacional Mayor de San Marcos |
| Repositorio: | Revistas - Universidad Nacional Mayor de San Marcos |
| Idioma: | español |
| OAI Identifier: | oai:revistasinvestigacion.unmsm.edu.pe:article/25686 |
| Acceso en línea: | https://revistasinvestigacion.unmsm.edu.pe/index.php/matema/article/view/25686 |
| Access Level: | acceso abierto |
| Palabra clave: | algebra of division conjugation quaternion octonion álgebra de división conjugación cuaternión octonión |
| Sumario: | 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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