Golden Ratio as Real Subfield of Cyclotomic Field ℚ(ζ₅) — E8 Intelligence Research

FINDING: Cyclotomic polynumbers (polynomials) define algebraic number fields; golden ratio φ = (1+√5) /2 generates the quadratic field ℚ (√5), which is the real subfield of the cyclotomic field ℚ (ζ₅) of 5th roots of unity. | MATH: φ = (1+√5) /2 = 2cos (π/5) ; ζ₅ = e^2πi/5; ℚ (ζ₅) has degree φ (5) =4; real subfield ℚ (ζ₅+ζ₅⁻¹) = ℚ (√5) = ℚ (φ) ; φ satisfies φ² = φ + 1; φ⁻¹ = φ - 1 = 0. 618…; φ² = φ + 1 = 2. 618…; φ⁻² = 2 - φ = 0. 382…; Moebius function μ (n) and Euler totient φ (n) appear in identities linking φ to cyclotomic polynomials: Φₙ (φ) = 0 for n=5; more generally, identities involve sums over divisors of n of μ (d) φ (n/d) etc. | CONNECTION: φ and its reciprocal 0. 618 are the fundamental ratios of pentagonal symmetry (crystallographic point group 5m) ; the 5-cycle symmetry of ζ₅ generates the golden ratio via 2cos (π/5) ; this is the same ratio appearing in quasicrystal diffraction patterns (Penrose tilings, icosahedral symmetry). The base-60 (sexagesimal) system is not directly present, but the pe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com