A crossed product of a skew field of quaternions and four-group

  • Valery V. Kursov Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Abstract

The article considers construction of generalized crossed product of an arbitrary quaternions skew field and Klein four-group relative to factor system. It is well known that such crossed product is semisimple ring. Under specific conditions it is easy to show that crossed product of simple algebra and its inner automorphism group is central simple algebra. Finding out the conditions under which the crossed product is division algebra we face to difficulties of general case linked to analysis of system of linear equations defined over non-commutative rings. In the terms of anisotropic quadratic forms, there are sufficient conditions under those the crossed product of skew field of quaternions and four-group relative to factor system is division algebra. In addition, it is proved that the crossed product is the tensor product of two quaternions skew fields. 

Author Biography

Valery V. Kursov, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

PhD (physics and mathematics), docent; associate professor at the department of higher algebra and information security, deputy dean of the faculty of mechanics and mathematics

References

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Published
2018-01-23
Keywords: quaternion, quaternions skew field, Klein four-group, crossed product, algebra, associative algebra, simple algebra, division algebra, factor system, tensor product
How to Cite
Kursov, V. V. (2018). A crossed product of a skew field of quaternions and four-group. Journal of the Belarusian State University. Mathematics and Informatics, 2, 12-16. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/743
Issue
No 2 (2017): Journal of the Belarusian State University. Mathematics and Informatics
Section
Mathematical Logic, Algebra and Number Theory

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