The Cayley-Dickson process

Let F be a field. A Dickson-algebra over F consists of a algebra K over F equipped with an involutorial anti-automorphism called conjugation, such that and for each element a in K. Note that

for a in K, i.e. K is a quadratic algebra over F.

We are going to define the Cayley-Dickson construction. Given a Dickson-algebra K over F and an element we have the vector space over F, and define a multiplication in L by the formula

The conjugation is defined to be

Then L becomes another Dickson-algebra with tr(u,v)=tr(u) and The algebra L is called a Cayley-Dickson extension of K. Note that L contains K as a subalgebra via the embedding

Theorem. Let K be a Dickson-algebra over a field F, and let L be a Cayley-Dickson extension of K, as above.

• L is commutative if and only if K is commutative and for all elements a of K.
• L is associative if and only if K is associative and commutative.
• L is a field if and only if K is a field, and for all u,v in K with v distinct from zero.
• L is a skewfield if and only if K is a field and for all u,v in K with v distinct from zero.
• L is an alternative field if and only if K is a skewfield and for all u,v in K with v distinct from zero.