composition algebras over finite fields
Theorem 1.
There are 5 nonisomorphic composition algebras^{} over a finite field^{} $k$ of characteristic not 2, 2 division algebras^{} and 3 split algebras^{}.

1.
The field $k$.

2.
The unique quadratic extension field $K/k$.

3.
The exchange algebra: $k\oplus k$.

4.
$2\times 2$ matrices over $k$: ${M}_{2}(k)$.

5.
The split Cayley algebra^{}.
Proof.
Following Hurwitz’s theorem every composition algebra is given by the CayleyDickson construction and has dimension^{} 1,2, 4 or 8. Now we consider the possible nondegenerate quadratic forms of these dimensions.
Since every anisotropic 2 space corresponds to a quadratic field extension, and our field is finite, it follows that there is at most one anisotropic 2 subspace^{} of our quadratic form^{}. Therefore if $dimC>2$ then the quadratic form is isotropic and so the algebra is a split. Therefore in the CayleyDickson construction over a finite field there every quaternion algebra^{} is split, thus ${M}_{2}(k)$. To build the nonassociative division Cayley algebra of dimension 8 requires we start the CayleyDickson construction with a division ring which is not a field, and thus there are no Cayley division algebras over finite fields. ∎
This result also can be seen as a consequence of Wedderburn’s theorem that every finite division ring is a field. Likewise, a theorem of Artin and Zorn asserts that every finite alternative division ring is in fact associative, thus excluding the Cayley algebras in a fashion similar to how Wedderburn’s theorem excludes division quaternion algebras.
Title  composition algebras over finite fields 

Canonical name  CompositionAlgebrasOverFiniteFields 
Date of creation  20130322 17:18:26 
Last modified on  20130322 17:18:26 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  9 
Author  Algeboy (12884) 
Entry type  Theorem 
Classification  msc 17A75 
Related topic  HurwitzsTheorem 
Related topic  JacobsonsTheoremOnCompositionAlgebras 