ABSTRACT
This chapter proves Hurwitz’s celebrated theorem that the only algebras with such a composition law are the well-known ones in 1, 2, 4, and 8 dimensions (exactly, for the algebras with a multiplicative identity, and up to isotopy, if not). We shall suppose at first that our algebra does possess a two-sided identity element 1, so that 1x = x = x1 and as usual, x,y=x+y−x−y2. The chapter first deduces multiplication laws and proves three conjugation laws. The chapter proves Hurwitz’s celebrated theorem, that a composition algebra with identity on a real Euclidean space is one of these four algebras. It shows that this is true up to isotopy for algebras without identities. The chapter also presents two appendices dealing with Dickson doubling rules and fixing of a quaternion subalgebra.
