In this two-part note I’m going to prove that every finite division ring is a field. This result is called Wedderburn’s little theorem. The proof we are going to give is due to Ernst Witt and that’s probably the best proof available. But before getting into the proof, we need to know a little bit about cyclotomic polynomials.

**Notation**. For any integer we have the -th root of unity

**Definition**. The -th **cyclotomic polynomial** is defined by

**Lemma 1**. In particular,

*Proof*. We have But

and obviously Hence

**Corollary**. For every

*Proof*. By induction over There is nothing to prove if because Now let and suppose the for all Note that cyclotomic polynomials are all monic. Thus, by Lemma 1

for some monic polynomial Since is monic too, it follows from that

**Lemma 2**. If and then

*Proof*. By Lemma 1 we have

Therefore, again by Lemma 1, where

We will continue our discussion in part (2).