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Zbl 0669.17001
Kantor, I.L.; Solodovnikov, A.S.
(Shenitzer, A.)
Hypercomplex numbers. An elementary introduction to algebras. Transl. by A. Shenitzer. (English)
[B] New York etc.: Springer-Verlag. x, 169 p. DM 78.00 (1989). ISBN 0-387-96980-2

[For a review of the German translation (Teubner 1978) see Zbl 0395.17001.] \par The original Russian edition appeared in 1973 as a text-book, intended for students of science high schools. As a matter of fact one cannot expect that this book contains new material and further developments in the study of hypercomplex systems. Nevertheless it was worthwhile to translate it into English because of the lucid way in which the subject is presented by the Russian authors. \par The book is subdivided into three chapters. The first part deals with the introduction of complex numbers, quaternions and Cayley numbers, presented as hypercomplex number systems and in the third chapter, titled `The exceptional position of four algebras', it is shown that the number in the ``sum of squares'' can take on just the four values 1,2,4 and 8 to obtain all division algebras: the reals, the complex numbers, the quaternions and the Cayley algebras. \par The second part, of an auxiliary nature, is an elementary exposition of the fundamental concepts of linear algebra.
[A.H.Boers]

MSC 2000:: *17-01 Textbooks (nonassoc. rings and algebras)
15-01 Textbooks (linear algebra)
17A35 Division algebras
16Kxx Division rings and semi-simple Artin rings
17D05 Alternative rings
16P10 Finite associative rings and finite-dimensional algebras
30G35 Functions of hypercomplex variables and generalized variables

Keywords: finite dimensional algebras; composition of quadratic forms; Hurwitz theorem; Frobenius theorem; complex numbers; quaternions; Cayley numbers; hypercomplex number systems; division algebras; linear algebra

Citations: Zbl 0669.17002; Zbl 0395.17001

Cited in: Zbl 1147.17002 Zbl 0669.17002

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	Scientific prize winners of the ICM 2010
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