It is proved that in a unital alternative algebra A of characteristic ≠ 2, the associator (a, b, c) and the Kleinfeld function f(a, b, c, d) never assume the value 1 for any elements a, b, c, d ∈ A. Moreover, if A is nonassociative, then no commutator [a, b] can be equal to 1. As a consequence, there do not exist algebraically closed alternative algebras. The restriction on the characteristic is essential, as exemplified by the Cayley–Dickson algebra over a field of characteristic 2.
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I. P. Shestakov Supported by FAPESP (project No. 2014/09310-5) and by CNPq (project No. 303916/2014-1).
Translated from Algebra i Logika, Vol. 58, No. 4, pp. 479-485, July-August, 2019.
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Kleinfeld, E., Shestakov, I.P. Associators and Commutators in Alternative Algebras. Algebra Logic 58, 322–326 (2019). https://doi.org/10.1007/s10469-019-09553-z
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DOI: https://doi.org/10.1007/s10469-019-09553-z