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 ℤφ is the ring of integers of ℚ (√5), a subfield of ℚ (ζ₅). - HMMT problem uses ℤφ 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