Bruck-Kleinfeld theorem on alternative division rings

Name
The result is due to Bruck and Kleinfeld and is termed the Bruck-Kleinfeld theorem.

Statement
There are only two possibilities for an alternative division ring. It can be either of these:


 * 1) It is associative, and hence a division ring in the usual sense of the term.
 * 2) It is an octonion algebra arising from a Cayley-Dickson construction.

Applications

 * Every finite alternative division ring is a field