# Division algebra

 Division algebra affects the special fields mathematics is special case by algebra 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 [itex] D< /math> a not necessarily associative algebra is, in to everyone [itex] the A \ in D< /math> and to everyone [itex] b \ in D, b \ neq 0< /math> exactly[itex] x \ in D< /math> with the characteristic [itex] a=x \ b /math< cdot> existed. (Designates “·” the Vektormultiplikation in algebra.) one still demands for the avoidance of a Trivialität that [itex] 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 [itex] 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)

This result is well-known as sentence of Hurwitz (1898).

Example of a division algebra without one element with the two units [itex] e_1< /math> and [itex] e_2< /math>,by arbitrary real numbers to be multiplied can:

[itex] \ 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} [/itex]