For notations and the results we have already proved, see part (1).
Lemma 3. Let be integers. Then
Proof. By definition of it suffices to prove that for any -th root of unity So if then and thus
Wedderburn’s Little Theorem. (J. M. Wedderburn, 1905) Every finite division ring is a field.
Proof. Let be a finite division ring with the center Then is a (finite) field and is a finite dimensional vector space over So if and then If then and we are done. So we will assume that and we will get a contradiction. Let as usual, be the multiplicative group of Clearly is the center of Also, for any let be the centralizer of in Then is also a finite division ring and thus a finite dimensional vector space over Let Then It is clear that the centralizer of in is So the class equation of gives us
By Lemma 1 and Lemma 2 in part (1), divides both and So by Thus contradicting Lemma 3.