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Some examples of real division algebras. (English) Zbl 1084.17500

Summary: It is known, by Frobenius Theorem, that the only division associative algebras over \(\mathbb R\) are \(\mathbb R\), \(\mathbb C\), \(\mathbb H\). In 1958 Bott and Milnor showed that a finite-dimensional real division algebra can have only dimensions \(1,2,4,8\). The algebras \(\mathbb R\), \(\mathbb C\), \(\mathbb H\) and \(\mathbb O\), first, second and third are associative and the fourth is non-associative, are the only finite-dimensional alternative real division algebras. In [J. Algebra 67, 479-490 (1980; Zbl 0451.17001)], by S. Okubo and H. C. Myung is given a construction of division non-unitary non-alternative algebras over an arbitrary field \(K\) with \(\text{char\,}K\neq2\). In this paper we analyse a case when these algebras are isomorphic.

MSC:

17A35 Nonassociative division algebras
17D05 Alternative rings

Citations:

Zbl 0451.17001
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