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. As our field is algebraically closed the quadratic form has no anisotropic subspaces and is therefore the unique quadratic form of maximal Witt index. Following Hurwitz’s theorem we know the composition algebras come in dimensions 1,2,4, and 8 and arise by the Cayley-Dickson method. Thus we have the field itself and the three split composition algebras. ∎

Title	composition algebra over algebaically closed fields
Canonical name	CompositionAlgebraOverAlgebaicallyClosedFields
Date of creation	2013-03-22 17:18:22
Last modified on	2013-03-22 17:18:22
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	6
Author	Algeboy (12884)
Entry type	Theorem
Classification	msc 17A75
Related topic	HurwitzsTheorem
Related topic	JacobsonsTheoremOnCompositionAlgebras