Except for a generalization of the so-called Wedderburn principal theorem, the structure theory of alternative algebras over an arbitrary field is as complete as that for associative algebras. It is our purpose here to fill this one gap in the alternative theory.
The principal theorem.A non-associative algebra 31 of order n over an arbitrary field % is called alternative in casefor all a, x in 21. It is clear that associative algebras are alternative.The most famous examples of alternative algebras which are not associative are the so-called Cayley-Dickson algebras of order 8 over $. Let S be an algebra of order 2 over % which is either a separable quadratic field over 5 or the direct sum 5 ©3-There is one automorphism z->z of S (over %) which is not the identity automorphism. The associative algebra O = 3~\~US with elements for /JT^O in § is called a quaternion algebra. For q in the form (1), the correspondence for 77^0 in %, where q-»g is the involution (3) of Q. Most of our knowledge of alternative algebras is due to M. Zorn.