So we have proved that if is a field of characteristic zero and with then is a commutative domain and it is a free module of finite rank over What can be said about the field of fractions of ? The next theorem shows that has a very simple form.

**Theorem 2**. (Amitsur, 1957) Let be a field of characteristic zero and let with Let and be the field of fractions of and respectively. Then is an algebraic extension of and for some

*Proof*. First note that, by Theorem 1, is a commutative domain and hence its field of fractions exists. Now let and be as defined in the proof of Theorem 1. We proved in that theorem that for every there exists some such that

If in we choose then we will get So is algebraic over and thus is a subfield of Also shows that for all and thus Therefore and hence