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. 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. Every finite division ring is a field
3. There are no finite alternative division rings that arise as Cayley-Dickson algebras

Proof

Direct from the given facts

The proof follows directly by combining Facts (1)-(3).