Let be a primitive th root of unity and let . Prove that is a primitive th root of unity.

Certainly . Now if , we have , and so divides . So is the order of , and thus is a primitive th root of unity.

unnecessary lemmas. very sloppy. handwriting needs improvement.

Let be a primitive th root of unity and let . Prove that is a primitive th root of unity.

Certainly . Now if , we have , and so divides . So is the order of , and thus is a primitive th root of unity.

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.