We are now going to prove that is a commutative -algebra for all with Recall that by we mean the largest power of in First a simple lemma.

**Lemma 2**. Let be a field of characteristic zero and let with If is an inetger, then the set consisting of all elements of of degree at most is a finite dimensional -vector space.

*Proof*. It is clear that is a -vector space. The proof of finite dimensionality of is by induction over If then and there is nothing to prove. So suppose that and fix an element with Of course, if there is no such then and we are done by induction. Now, let If then and if then there exists some such that by Lemma 1. Thus and hence and we are done again by induction.

**Theorem 2**. (Amitsur, 1957) Let be a field of characteristic zero and let with Then is commutative.

*Proof*. Let and be as defined in the proof of Theorem 1. As we mentioned in there, is a cyclic subgroup of of order for some divisor of $n.$ Let be a generator of and choose such that Now let

Clearly Let So basically is the set of all non-negative integers which appear as the degree of some element of Let Then for some inetger because is a generator of Hence for some integer If then and if then Thus if and then Let be the set of all elements of of degree at most By Lemma 2, is -vector space and The claim is that

Clearly because both and are in To prove let We use inducton over If then and hence by Lemma 1. If then and we are done. Otherwise, and hence there exists some such that Thus, by Lemma 1, there exists some such that Therefore by induction and hence because This completes the proof of

Now let and let Clearly and hence

for some Since the elements are -linearly dependent and so for some which are not all zero. It now follows from that where So we have proved that for every there exists some such that Let and let be such that and Then, since is clearly commutative, we have Therefore, since is commutative and and commute with we have Thus, since is a domain and we have Hence is commutative.

In part (4), which will be the last part, we will find the field of fractions of