Golden Ratio φ as Algebraic Integer in Cyclotomic Field ℚ(ζ₅) Links Class Number Theory — E8 Intelligence Research

**FINDING:** The golden ratio φ is an algebraic integer in the cyclotomic field ℚ(ζ₅), linking class number theory of ℚ(ζ₅) to φ's arithmetic properties. **MATH:** - φ = (1 + √5)/2 satisfies φ² = φ + 1, minimal polynomial x² – x – 1. - ℚ(ζ₅) = ℚ(√5, √–5) has discriminant 5³, class number 1 (unique factorization). - φ = 2 cos(π/5) = ζ₅ + ζ₅⁻¹, where ζ₅ = e^{2πi/5}. - Ring ℤ[φ] is the ring of integers of ℚ(√5), a subfield of ℚ(ζ₅). - HMMT problem uses ℤ[φ] for algebraic number theory in competition. **CONNECTION:** - φ = 1.618…, its reciprocal 0.618…, and φ² = 2.618… are all present. - 2 cos(π/5) = φ relates to pentagonal symmetry (crystallographic point group 5m). - Base-60: φ approximates 1;36,48,20 in sexagesimal (1 + 36/60 + 48/3600 + …). - Class number 1 of ℚ(ζ₅) means the cyclotomic field has unique factorization, a rare property (only for n = 1,2,3,4,5,7,8,9,11,13,17,19,23,25,27,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97). - Pentagonal numbers and root s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com