Let be a finite extension of . Prove that contains only finitely many roots of unity.
Suppose to the contrary that contains infinitely many roots of unity. Now for each , there are only finitely many primitive roots of unity (in fact of them). So for each , the number of primitive roots of unity of order at most is finite. In particular, for any , there exists a primitive th root of unity for some .
Let be the degree of over . If is a primitive th root of unity, then by Corollary 42 on page 555, where denotes the Euler totient. By this previous exercise, since is arbitrarily large, is arbitrarily large. So there exists a primitive th root such that , a contradiction since .