 Octonion algebra

In mathematics, an octonion algebra over a field F is an algebraic structure which is an 8dimensional composition algebra over F. In other words, it is a unital nonassociative algebra A over F with a nondegenerate quadratic form N (called the norm form) such that
 N(xy) = N(x)N(y)
for all x and y in A.
The most wellknown example of an octonion algebra are the classical octonions, which are an octonion algebra over R, the field of real numbers. The splitoctonions also form an octonion algebra over R. Up to Ralgebra isomorphism, these are the only octonion algebras over the reals.
A split octonion algebra is one for which the quadratic form N is isotropic (i.e. there exists a vector x with N(x) = 0). Up to Falgebra isomorphism, there is a unique split octonion algebra over any field F. When F is algebraically closed or a finite field, these are the only octonion algebras over F.
Octonion algebras are always nonassociative. They are however alternative algebras (a weaker form of associativity). Moreover, the Moufang identities hold in any octonion algebra. It follows that the set of invertible elements in any octonion algebra form a Moufang loop, as do the subset of unit norm elements.
Classification
It is a theorem of Adolf Hurwitz that the Fisomorphism classes of the norm form are in onetoone correspondence with the isomorphism classes of octonion Falgebras. Moreover, the possible norm forms are exactly the Pfister 3forms over F.
Since any two octonion Falgebras become isomorphic over the algebraic closure of F, one can apply the ideas of nonabelian Galois cohomology. In particular, by using the fact that the automorphism group of the split octonions is the split algebraic group G_{2}, one sees the correspondence of isomorphism classes of octonion Falgebras with isomorphism classes of G_{2}torsors over F. These isomorphism classes form the nonabelian Galois cohomology set H^{1}(F,G_{2}).
See also
References
 Springer, T. A.; F. D. Veldkamp (2000). Octonions, Jordan Algebras and Exceptional Groups. SpringerVerlag. ISBN 3540663371.
 Serre, J. P. (2002). Galois Cohomology. SpringerVerlag.
Categories: Octonions
 Nonassociative algebras
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