You can see part (1) in here.

**Wedderburn’s Factorization Theorem**. (Wedderburn, 1920) Let be a division algebra with the center Suppose that is algebraic over and let be the minimal polynomial of over There exist non-zero elements such that

*Proof*. By Remark 2 in part (1), there exists such that Now let be the largest integer for which there exist non-zero elements and such that

Let

*Claim*. for all

*Proof of the claim*. Suppose, to the contrary, that there exists such that Then, since and there exists such that by Lemma 1 in part (1). So, by Remark 2 in part (1), there exists such that Hence contradicting the maximality of

Therefore, by Lemma 2 in part (1), and so because and both and are monic.

In the next post, I will use Wedderburn’s factorization theorem to find an expression for the reduced trace and the reduced norm of an element in a finite dimensional central division algebra.