Let be an odd integer. Prove that , where denotes the th cyclotomic polynomial.
We begin with a lemma.
Lemma: -1 is not an th root of unity for odd . Proof: .
Let be a primitive th root of unity. Now suppose , so that , so . Since is a primitive th root of unity, with odd, the powers of are th roots of unity. By the lemma, we must have , so that , and thus , so that . Since and are relatively prime, , and so is a primitive th root of unity.
In particular, divides .
Now has degree , and has degree , where denotes the Euler totient. Since and are relatively prime, we have . So in fact .