composition algebras over $\mathbb{R}$

There are 7 non-isomorphic composition algebras over $\mathbb{R}$ , first 4 division algebras and secondly 3 split algebras.

1.

The real numbers $\mathbb{R}$ .
2.

The complex numbers $\mathbb{C}$ .
3.

The Hamiltonians (also known as the quaternions) $\mathbb{H}$ .
4.

The octonions (also known as the Cayley or Cayley-Dickson algebra) $\mathbb{O}$ .
5.

The exchange algebra: $\mathbb{R}\oplus\mathbb{R}$ .
6.

$2\times 2$ matrices over $\mathbb{R}$ : $M_{2}(\mathbb{R})$ .
7.

The cross-product of $2\times 2$ -matrices over $\mathbb{R}$ : $M_{2}(\mathbb{R})\circ M_{2}(\mathbb{R})$ .

The proof can be seen as a consquence of a theorem of Hurwitz and a theorem of Jacobson. In reality various authors contributed to the solution including Albert, Dickson and Kaplansky.

Title	composition algebras over $\mathbb{R}$
Canonical name	CompositionAlgebrasOvermathbbR
Date of creation	2013-03-22 17:18:17
Last modified on	2013-03-22 17:18:17
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	7
Author	Algeboy (12884)
Entry type	Example
Classification	msc 17A75
Related topic	HurwitzsTheorem
Related topic	JacobsonsTheoremOnCompositionAlgebras

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composition algebras over $\mathbb{R}$