Every finite division ring is a field

History
This result was first proved by Wedderburn.

Statement
Every finite division ring is a  field, and hence, a  finite field. Here, by division ring, we mean associative division ring.

Stronger facts

 * Bruck-Kleinfeld theorem states that every alternative division ring is either associative (and hence a division ring in the usual sense) or is a Cayley-Dickson algebra.
 * In particular, this shows that every finite alternative division ring is a field.

Facts used

 * 1) uses::Lagrange's theorem
 * 2) uses::Class equation of a group

Proof
Given: A division ring $$K$$ of finite size.

To prove: $$K$$ is a field.

Proof': We denote by $$K^*$$ the multiplicative group of nonzero elements of $$K$$.

The key idea behind the proof is to switch between the additive/linear structure and the multiplicative structure. For the additive structure, we use key fact that the size of a vector space over a field is a power of the size of the field. For the multiplicative structure, we use Lagrange's theorem (a multiplicative constraint) and the class equation (a combinatorial constraint on how things should add up). Finally, we show that the number theory does not work out.