Hypercomplex Numbers and Roots of Algebraic Equation

IF 0.5 Q4 PHYSICS, MATHEMATICAL

Journal of Geometry and Symmetry in Physics Pub Date : 2022-01-01 DOI:10.7546/jgsp-64-2022-9-22

Ying-Qiu Gu

引用次数: 0

Abstract

By means of hypercomplex numbers, in this paper we discuss algebraic equations and obtain some interesting relations. A structure equation $A^2=nA$ of a group is derived. The matrix representation of a group constitutes the basis elements of a hypercomplex number system. By a canonical real matrix representation of a cyclic group, we define the cyclic number system, which is exactly the solution space of the higher order algebraic equations, and thus can be used to solve the roots of algebraic equations. Hypercomplex numbers are linear algebras with definition of multiplication and division, satisfying the associativity and distributive law, which provide a unified, standard, and elegant language for many complex mathematical and physical objects. So, we have one more proof that the hypercomplex numbers are worthy of application in teaching and scientific research.

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超复数与代数方程的根

本文利用超复数讨论了代数方程，得到了一些有趣的关系。导出了群的结构方程A^2=nA$。群的矩阵表示构成了超复数系统的基元。通过循环群的正则实矩阵表示，定义了循环数系统，该系统正是高阶代数方程的解空间，因而可用于求解代数方程的根。超复数是具有乘法和除法定义、满足结合律和分配律的线性代数，它为许多复杂的数学和物理对象提供了一种统一、标准和优雅的语言。由此，我们又一次证明了超复数在教学和科研中的应用价值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Symmetry in Physics PHYSICS, MATHEMATICAL-

CiteScore

1.50

自引率

25.00%

发文量

期刊介绍： The Journal of Geometry and Symmetry in Physics is a fully-refereed, independent international journal. It aims to facilitate the rapid dissemination, at low cost, of original research articles reporting interesting and potentially important ideas, and invited review articles providing background, perspectives, and useful sources of reference material. In addition to such contributions, the journal welcomes extended versions of talks in the area of geometry of classical and quantum systems delivered at the annual conferences on Geometry, Integrability and Quantization in Bulgaria. An overall idea is to provide a forum for an exchange of information, ideas and inspiration and further development of the international collaboration. The potential authors are kindly invited to submit their papers for consideraion in this Journal either to one of the Associate Editors listed below or to someone of the Editors of the Proceedings series whose expertise covers the research topic, and with whom the author can communicate effectively, or directly to the JGSP Editorial Office at the address given below. More details regarding submission of papers can be found by clicking on "Notes for Authors" button above. The publication program foresees four quarterly issues per year of approximately 128 pages each.