Suppose and are relatively prime, and let and be primitive th and th roots of unity, respectively. Show that is a primitive th root of unity.

Note first that , so that is an th root of unity.

Now let be the order of ; we have , so that . In particular, and have the same order, which must be a divisor of both and . Since and are relatively prime, the order of is 1, so . Likewise . So and , and again since and are relatively prime, . So , and is a primitive th root of unity.

Prove that for any prime and any nonzero , is irreducible and separable over .

Note that , so that and are relatively prime. So is separable.

Now let be a root of . Using the Frobenius endomorphism, , so that is also a root. By induction, is a root for all , and since has degree , these are all of the roots.

Now , and in particular the minimal polynomials of and have the same degree over – say . Since is the product of the minimal polynomials of its roots, we have for some . Since is prime, we have either (so that , a contradiction) or , so that itself is the minimal polynommial of , hence is irreducible.

Fix an integer . Prove that for all , divides if and only if divides . Conclude that if and only if .

If , then by this previous exercise, divides in , and so divides in .

Conversely, suppose divides . Using the Division Algorithm, say . Now . If , then we have , so that and . If , then , and again we have , so that as desired.

Recall that is precisely the roots (in some splitting field) of . Now if and only if divides by the above argument, if and only if divides by a previous exercise, if and only if divides , if and only if .