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 constructionand has dimension 1,2, 4 or 8. Now we consider the possible non-degeneratequadratic 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. Thereforeif $\\dim C>2$ 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 quaternionalgebra is split, thus $M_{2}(k)$ . To build the non-associative division Cayley algebra of dimension 8 requireswe start the Cayley-Dickson construction with a division ring which is not a field, and thus there are noCayley division algebras over finite fields.∎

This result also can be seen as a consequence of Wedderburn’s theorem that every finite division ringis 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 divisionquaternion algebras.

单词	CompositionAlgebrasOverFiniteFields
释义	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 constructionand has dimension 1,2, 4 or 8. Now we consider the possible non-degeneratequadratic 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. Thereforeif $\\dim C>2$ 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 quaternionalgebra is split, thus $M_{2}(k)$ . To build the non-associative division Cayley algebra of dimension 8 requireswe start the Cayley-Dickson construction with a division ring which is not a field, and thus there are noCayley division algebras over finite fields.∎ This result also can be seen as a consequence of Wedderburn’s theorem that every finite division ringis 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 divisionquaternion algebras.
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