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Minimal identities of octonion algebras. (English) Zbl 0651.17012

Let \(K\) be a field of a characteristic not \(2, 3\) or \(5\). The main theorem in the paper is an analog of the Amitsur-Levitzki theorem namely: There are no identities of degrees less than 5 for the octonion algebras over \(K\). The author has proved that all identities of degree \(5\) for the octonion algebras follow from the identities \([[x,y]^ 2, z]=0\) and \(x^ 2 S^ +_ 3(y,z,w)-x S^ +_ 3(y,z,w)\circ x=0\) where \(S^ +_ n(x_ 1,...,x_ n)\) is the Jordan standard identity.

MSC:

17D05 Alternative rings
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17A50 Free nonassociative algebras
17A45 Quadratic algebras (but not quadratic Jordan algebras)
17C05 Identities and free Jordan structures
17A60 Structure theory for nonassociative algebras
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References:

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