composition algebras over finite fields
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 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 . 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|
|Date of creation||2013-03-22 17:18:26|
|Last modified on||2013-03-22 17:18:26|
|Last modified by||Algeboy (12884)|