Fourier Formula for Cyclotomic Polynomials at Non-Primitive Roots — E8 Intelligence Research

FINDING: Cyclotomic polynomials Φₙ(x) are the minimal integer polynomials for primitive n-th roots of unity; a novel Fourier-analytic formula evaluates Φₙ at non-primitive roots, linking root systems to discrete harmonic analysis. MATH: Φₙ(x) = ∏1≤k≤n, gcd(k,n)=1 (x − ζₙᵏ), ζₙ = e2πi/n. Degree = φ(n). Key identity: xⁿ − 1 = ∏d|n Φ_d(x). The arXiv result (1611.06783) gives Φₙ(ζₘ) for m∤n via finite Fourier sums: Φₙ(ζₘ) = exp( Σd|n μ(n/d) · log(1 − ζₘᵈ) ) — a Möbius-inverted trace over divisors. For m|n, Φₙ(ζₘ)=0 trivially. CONNECTION: The roots ζₙᵏ form a regular n-gon on the unit circle — a cyclic lattice with rotational symmetry Cₙ. For n=5, the primitive roots give the golden ratio: 2cos(2π/5) = (√5−1)/2 ≈ 0.618; 2cos(π/5) = φ ≈ 1.618. For n=10, 2cos(π/10) = √(2.618) ≈ 1.618. The Fourier formula exposes a multiplicative structure over divisors — a discrete analogue of the zeta function's Euler product, echoing base-60's divisor-rich arithmetic (60 has 12 divisors, ena Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com