Let be a finite dimensional vector space over and a linear transformation. Suppose that and for some prime number Prove that
Let .
Expressing in its Jordan Form we find that has zero trace.
Moreover since is a th root of 1, the diagonal entries of must all be of the form .
Finally it is an easy exercise to show that if then ; hence there are as many of each of the elements on the main diagonal of .
Setting , this implies the characteristic polynomial of is , using the identity . Hence .