# composition algebras over finite fields

###### Theorem 1.

1. 1.

The field $k$.

2. 2.

The unique quadratic extension field $K/k$.

3. 3.

The exchange algebra: $k\oplus k$.

4. 4.

$2\times 2$ matrices over $k$: $M_{2}(k)$.

5. 5.
###### Proof.

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 $\dim C>2$ then the quadratic form is isotropic and so the algebra is a split. Therefore in the Cayley-Dickson construction over a finite field there every quaternion algebra  is split, thus $M_{2}(k)$. To build the non-associative division Cayley algebra of dimension 8 requires we start the Cayley-Dickson 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 CompositionAlgebrasOverFiniteFields 2013-03-22 17:18:26 2013-03-22 17:18:26 Algeboy (12884) Algeboy (12884) 9 Algeboy (12884) Theorem msc 17A75 HurwitzsTheorem JacobsonsTheoremOnCompositionAlgebras