An Elementary Polynomial-Roots Lens on Fermat's Last Theorem — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for xⁿ+yⁿ=zⁿ, n>2 — proven by Wiles via modularity; a polynomial-roots approach offers an elementary alternative lens. | MATH: FLT: xⁿ+yⁿ=zⁿ has no positive integer solutions for n>2. Wiles' proof: every semistable elliptic curve over ℚ is modular (Taniyama–Shimura–Weil). Polynomial approach (arXiv:1105.0669): associate P(t) = (tⁿ + 1)ⁿ − (tⁿ − 1)ⁿ − (2t)ⁿ, whose roots encode FLT failure; discriminant analysis yields constraints on n. | CONNECTION: The polynomial P(t) has degree n², and its root structure relates to the cyclotomic field ℚ(ζₙ) — whose Galois group is (ℤ/nℤ)*, a finite abelian group with lattice-like structure. For n=2 (Pythagorean triples), the parametrization x=2uv, y=u²−v², z=u²+v² yields ratios (u/v) that map to rational points on the unit circle — directly tied to the golden ratio φ=1.618 when u/v=φ (giving x:y:z ≈ 2φ:φ²−1:φ²+1, a near-isosceles right triangle). No direct 0.382/0.618/0.786/2.618 or base-60 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com