**Theorem 1**. (Amitsur, 1957) Let be a field of characteristic zero and let with Then is a free -module of rank where is a divisor of

*Proof*. Suppose that is the set of all integers for which there esists some such that Clearly is a submonoid of For any let be the image of in and put Since is a submonoid of a finite cyclic group, it is a cyclic subgroup and hence divides Let where and, in general, each is chosen to be non-negative and the smallest member of its class That means if and then For any let with So we can choose to be any element of degree zero in We choose To complete the proof of the theorem, we are going to show that, as a -module, generate and are linearly independent over We first show that Clearly because for all Now let and suppose that If then and hence, by Lemma 1, If then for some We also have by minimality of Thus for some integer Therefore Now both and are obviously in So if and are the leading coefficeints of and respectively, then by Lemma 1, for some Therefore and, since we can apply induction on to get Thus It remains to show that are linearly independent over Suppose, to the contrary, that

for some and not all are zero. Note that if and then and Since we have and hence Thus the left hand side of is a polynomial of degree and so it cannot be equal to zero.

In part (3) we will prove that is commutative.