Let , where is a prime not dividing , and let be a field of characteristic . Prove that there are distinct th roots of unity over .

Note that . That is, the distinct th roots of unity over are precisely the distinct roots of over .

If , then certainly there is only 1 root of .

Suppose . Now , which has only the root 0 with multiplicity . Clearly 0 is not a root of , so that and its derivative are relatively prime, and thus is separable. Hence there are distinct th roots of unity over , and so distinct th roots of unity over .