Abstract
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.
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The author would like to thank the Public Authority for Applied Education and Training, for supporting and funding this research. Project No BE-14-06.
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To whom correspondence should be M. Bani-Ata.
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Bani-Ata, M., Al-Rashed, M. On certain finite dimensional algebras over finite fields. Beitr Algebra Geom 58, 195–200 (2017). https://doi.org/10.1007/s13366-016-0312-8
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DOI: https://doi.org/10.1007/s13366-016-0312-8