composition algebras over finite fields

Theorem 1.

There are 5 non-isomorphic composition algebrasMathworldPlanetmath over a finite fieldMathworldPlanetmath k of characteristic not 2, 2 division algebrasMathworldPlanetmath and 3 split algebrasPlanetmathPlanetmath.

  1. 1.

    The field k.

  2. 2.

    The unique quadratic extension field K/k.

  3. 3.

    The exchange algebra: kk.

  4. 4.

    2×2 matrices over k: M2(k).

  5. 5.

    The split Cayley algebraMathworldPlanetmathPlanetmath.


Following Hurwitz’s theorem every composition algebra is given by the Cayley-Dickson construction and has dimensionPlanetmathPlanetmathPlanetmath 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 subspacePlanetmathPlanetmathPlanetmath of our quadratic formMathworldPlanetmath. 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 algebraPlanetmathPlanetmath is split, thus M2(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