composition algebras over $\\mathbb{R}$

There are 7 non-isomorphic composition algebras over $\\mathbb{R}$ , first 4 division algebras and secondly3 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 andKaplansky.

单词	CompositionAlgebrasOvermathbbR
释义	composition algebras over $\\mathbb{R}$ There are 7 non-isomorphic composition algebras over $\\mathbb{R}$ , first 4 division algebras and secondly3 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 andKaplansky.
随便看	PerfectCode perfect code PerfectField perfect field PerfectGroup perfect group PerfectNumber perfect number PerfectPower perfect power PerfectRuler perfect ruler PerfectSet perfect set PerfectTotientNumber perfect totient number PerimeterOfAstroid perimeter of astroid Period period PeriodicContinuedFractionsRepresentQuadraticIrrationals periodic continued fractions represent quadratic irrationals PeriodicFunctions periodic functions PeriodicGroup

composition algebras over ℝ

composition algebras over $\\mathbb{R}$