On Cyclotomic Polynomials with ± 1 Coefficients

We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set {+1,−1}. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd coefficients. The characterization is as follows. A polynomial P (x) with coefficients ±1 of even degree N − 1 is cyclotomic if and only if P (x) = ±Φp1(±x)Φp2(±x) · · ·Φpr (±xp1p2···pr−1), where N = p1p2 · · · pr and the pi are primes, not necessarily distinct. Here Φp(x) := x−1 x−1 is the pth cyclotomic polynomial. We conjecture that this characterization also holds for polynomials of odd degree with ±1 coefficients. This conjecture is based on substantial computation plus a number of special cases. Central to this paper is a careful analysis of the effect of Graeffe’s root squaring algorithm on cyclotomic polynomials.