Octonion Algebra - Classification

Classification

It is a theorem of Adolf Hurwitz that the F-isomorphism classes of the norm form are in one-to-one correspondence with the isomorphism classes of octonion F-algebras. Moreover, the possible norm forms are exactly the Pfister 3-forms over F.

Since any two octonion F-algebras become isomorphic over the algebraic closure of F, one can apply the ideas of non-abelian Galois cohomology. In particular, by using the fact that the automorphism group of the split octonions is the split algebraic group G₂, one sees the correspondence of isomorphism classes of octonion F-algebras with isomorphism classes of G₂-torsors over F. These isomorphism classes form the non-abelian Galois cohomology set .