Let F be a field. We will construct the generalized quaternions
and the generalized octonions over F, respectively. This
construction will depend on the three parameters a,b,c in
the field F.

First, let a be an element of F with
for each u in F. Then the multiplication

turns
into a field. This field becomes a
Dickson-algebra using the
conjugation
Of course,
has dimension 2 over F.

Now, let b be another element in F with
for all elements u_{1}, u_{2} in F. Let
be the Cayley-Dickson extension of
with respect to the given parameter b. Then
becomes a non-commutative skewfield, with F as its center. This
skewfield is called a generalized quaternion algebra over F, obviously
it is of dimension 4 over F.

Finally, let c be another parameter in F with
for all u_{1}, u_{2}, u_{3}, u_{4}
in F. The Cayley-Dickson extension
of
with respect to the parameter c is called a generalized octionon
algebra over F. Note that
is a proper alternative field
(i.e.
is not associative,) of dimension 8 over

The classical quaternions and octonions are obtained by using
the field of real numbers as our base field F with the parameters
a=b=c=-1.

### See also

Contributed by Hauke Klein

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