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. 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 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.