# 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.

### See also

Contributed by Hauke Klein

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