composition algebra over algebaically closed fields
Theorem 1.
There are 4 nonisomorphic 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 crossproduct 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 2dimensional 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 CayleyDickson 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  20130322 17:18:22 
Last modified on  20130322 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 