On certain finite dimensional algebras over finite fields

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• 作者：Mashhour Bani-Ata ; Mariam Al-Rashed
• 关键词：Structure constant maps ; Linearized polynomials ; Homogeneous polynomials
• 刊名：Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：58
• 期：1
• 页码：195-200
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algebra; Geometry; Algebraic Geometry; Convex and Discrete Geometry;
• 出版者：Springer Berlin Heidelberg
• ISSN：2191-0383
• 卷排序：58

文摘

This paper is divided into four sections as follows: Sect. 1 is an introduction. In Sect. 2, we investigate the relation between \(\mathbb {F}_q\)-algebras arising from structure-constant maps and \(\mathbb {F}_q\)-algebras admitting elementary abelian groups as automorphism subgroups. In Sect. 3, we show that the elements of \(End_{K}(A)\), A is an algebra over a field K, can be expressed as linearized polynomials. In the last section, we study division algebras and the associated homogeneous polynomials over the finite field \(\mathbb {F}_q\) using tools from algebraic geometry and in particular Chevalley–Warning’s Theorem.