Every finite alternative division ring is a field

Statement
Any alternative division ring that is finite (i.e., its underlying set is finite) must be a field.

Facts used

 * 1) uses::Bruck-Kleinfeld theorem on alternative division rings: This says that every alternative division ring is either a division ring or a Cayley-Dickson algebra.
 * 2) uses::Every finite division ring is a field
 * 3) There are no finite alternative division rings that arise as Cayley-Dickson algebras

Direct from the given facts
The proof follows directly by combining Facts (1)-(3).