composition algebra over algebaically closed fields

Theorem 1.

There are 4 non-isomorphic composition algebras over an algebraically closed field $k$ :one division algebra, the field itself, and the three split algebras.

1.
$k$ .
2.
The exchange algebra: $k\\oplus k$ .
3.
$2\\times 2$ matrices over $k$ : $M_{2}(k)$ .
4.
The cross-product of $2\\times 2$ -matrices over $k$ : $M_{2}(k)\\circ M_{2}(k)$ .

Proof.

To see this recall that every composition algebra comes equipped with a quadratic form.Any 2-dimensional anisotropic subspace arises from a quadratic field extension. Asour field is algebraically closed the quadratic form has no anisotropic subspaces andis therefore the unique quadratic form of maximal Witt index. Following Hurwitz’s theoremwe know the composition algebras come in dimensions 1,2,4, and 8 and arise by theCayley-Dickson method. Thus we have the field itself and the three splitcomposition algebras.∎

单词	CompositionAlgebraOverAlgebaicallyClosedFields
释义	composition algebra over algebaically closed fields Theorem 1. There are 4 non-isomorphic composition algebras over an algebraically closed field $k$ :one division algebra, the field itself, and the three split algebras. 1. $k$ . 2. The exchange algebra: $k\\oplus k$ . 3. $2\\times 2$ matrices over $k$ : $M_{2}(k)$ . 4. The cross-product of $2\\times 2$ -matrices over $k$ : $M_{2}(k)\\circ M_{2}(k)$ . Proof. To see this recall that every composition algebra comes equipped with a quadratic form.Any 2-dimensional anisotropic subspace arises from a quadratic field extension. Asour field is algebraically closed the quadratic form has no anisotropic subspaces andis therefore the unique quadratic form of maximal Witt index. Following Hurwitz’s theoremwe know the composition algebras come in dimensions 1,2,4, and 8 and arise by theCayley-Dickson method. Thus we have the field itself and the three splitcomposition algebras.∎
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