# composition algebras over finite fields

###### Theorem 1.

There are 5 non-isomorphic composition algebras over a finite field $k$ of characteristic not 2, 2 division algebras and 3 split algebras.

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.

The split Cayley algebra.

###### Proof.

Following Hurwitz’s theorem every composition algebra is given by the Cayley-Dickson construction and has dimension 1,2, 4 or 8. Now we consider the possible non-degenerate 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 $\dim C>2$ 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