Fourier Formula Unveils Hidden Structure in Cyclotomic Polynomials — E8 Intelligence Research

FINDING: Cyclotomic polynomials Φₙ(x) are the minimal integer polynomials of primitive n-th roots of unity; a Fourier-analytic formula evaluates them at non-primitive roots, revealing hidden multiplicative structure. | MATH: Φₙ(x) = ∏_{1≤k≤n, gcd(k,n)=1} (x − ζₙᵏ), ζₙ = e^{2πi/n}; deg Φₙ = φ(n); Φₙ(x) = ∏_{d|n} (x^d − 1)^{μ(n/d)} (Möbius inversion); for m|n, Φₙ(ζₘ) = p^{φ(n)/φ(m)} if n/m is a prime power p^a, else 0 or ±1 (from arXiv 1611.06783). | CONNECTION: Roots of unity form the cyclic group μₙ — a 1D lattice on the unit circle. The Fourier formula for Φₙ at roots of unity uses discrete Fourier transforms, whose eigenbasis is exactly the characters of Z/nZ — a cyclic lattice. The prime-power case yields p^{φ(n)/φ(m)} — a power of a prime, echoing base-p (and base-60's prime factors 2,3,5) multiplicative structure. The symmetry group Gal(Q(ζₙ)/Q) ≅ (Z/nZ)^× has order φ(n), whose divisors relate to crystallographic point groups for n = 2,3,4,6 (hexagonal, tetragonal, trigonal). No d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com