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. 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 $dimC>2$ 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
Canonical name	CompositionAlgebrasOverFiniteFields
Date of creation	2013-03-22 17:18:26
Last modified on	2013-03-22 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