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 H^1(F, G_2). ==References==