# 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. 1.

The real numbers $\mathbb{R}$.

2. 2.

The complex numbers $\mathbb{C}$.

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 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 $\mathbb{R}$ CompositionAlgebrasOvermathbbR 2013-03-22 17:18:17 2013-03-22 17:18:17 Algeboy (12884) Algeboy (12884) 7 Algeboy (12884) Example msc 17A75 HurwitzsTheorem JacobsonsTheoremOnCompositionAlgebras