Let p be prime, and let , w is a primitive pth root of unity. Compute and
Also, show that
proof.
So I have with
If is a finite Galois extension then we define for . Now if is a primitive -th root of unity then is finite and Galois over , furthermore, since for a basis for this extension and is a field with it means is a group of order , so it must be a cyclic group. Thus, there is an automorphism of leaving fixed so that . Suppose that where . Then it means . And so . Thus, .