310 On the Coefficients of Cyclotomic Polynomials

Cyclotomy is the process of dividing a circle into equal parts, which is precisely the effect obtained by plotting the n-th roots of unity in the complex plane. For integers n ≥ 1, we know that Xn−1 = ∏n−1m=0(X−e 2piimn) over C. The n-th cyclotomic polynomial can be defined as Φn(X) = n∏ m=1,(m,n)=1 (X − e 2piimn) where e 2pii n is a primitive n-root of unity. Clearly, the degree of Φn(X) is φ(n) where φ is the Euler totient function. We have Xn − 1 = ∏d|n Φd(X). Lemma 1.1 The cyclotomic polynomial Φn(X) is a monic polynomial over integers. Proof. We use induction to prove this result. We have Φ1(X) = X − 1. We assume that the result is true for all d < n and we prove the result for n. By the induction hypothesis, we have F (X) def= d<n,d|n Φd(X) ∈ Z[X] and its leading coefficient is 1. As F (X) is monic, by division algorithm, ∃ h(X), r(X) ∈ Z[X] such that h(X) is monic and Xn − 1 = F (X)h(X) + r(X), where r(X) = 0 or degr(X) < degF (X). But, Xn − 1 = F (X)Φn(X). Therefore, by uniqueness of quotient and remainder in C[X], we must have h(X) = Φn(X). Also it is clear that Φn(X) has leading coefficient 1. The Möbius function, µ(n), is defined by µ(n) = 1 if n = 1, (−1)k if n = p1p2 · · · pk for distinct primes pi, 0 otherwise. Note that it can be easily seen that µ(mn) = µ(m)µ(n) whenever (m,n) = 1. Also, d|n µ(d) = 1 if n = 1, 0 otherwise Lemma 1.2 If µ(n) denotes the Möbius function, then, Φn(X) = d|n (Xn/d − 1)µ(d) = d|