Division algebra
Division algebra |
affects the special fields |
is special case by |
enclosure as special cases |
division algebra is a term from the mathematical subsection abstract algebra. Roughly spoken it concerns with a division algebra oneVector space, in which one can multiply and divide elements.
definition and examples
a division algebra <math> D< /math> a not necessarily associative algebra is, in to everyone <math> the A \ in D< /math> and to everyone <math> b \ in D, b \ neq 0< /math> exactly<math> x \ in D< /math> with the characteristic <math> a=x \ b /math< cdot> existed. (Designates “·” the Vektormultiplikation in algebra.) one still demands for the avoidance of a Trivialität that <math> D< /math> contains at least two elements.
A division algebra over the real numbers has always the dimension 1,2, 4 or 8. That was proven 1958 by Milnor and Kervaire.
Division algebra contains the number of 1, so that <math> a*1 = 1*a = A< /math> applies,one speaks of a division algebra with unity.
4 real division algebras with unity are (up to isomorphism)
- the real numbers themselves
- the complex numbers
- the quaternions
- the Oktaven also Oktonionen or Cayley numbers.
This result is well-known as sentence of Hurwitz (1898).
Example of a division algebra without one element with the two units <math> e_1< /math> and <math> e_2< /math>,by arbitrary real numbers to be multiplied can:
<math> \ begin {matrix} e_1 * e_1 &=& e_1 \ \ e_1 * e_2 &=& - e_2 \ \ e_2 * e_1 &=& - e_2 \ \ e_2 * e_2 &=& - e_1 \ end {to matrix} </math>