Galois Groups of Powered Golden Ratio Polynomials: The Splitting Field of x^{2ℓ} - x^ℓ - 1

Abstract

This preprint gives a complete arithmetic and Galois-theoretic analysis of the powered golden-ratio polynomial family Pℓ(x) = x^(2ℓ) − x^ℓ − 1, ℓ ≥ 2, whose roots are controlled by ℓ-th roots of the golden ratio φ = (1 + √5)/2 and its conjugate −φ⁻¹. For every integer ℓ ≥ 2, the polynomial Pℓ is proved irreducible over ℚ, and its splitting field is determined explicitly in terms of cyclotomic and Kummer extensions of ℚ(√5). The interaction between the quadratic conductor 5 and the cyclotomic field produces an exact distinction according to the parity of ℓ and whether 5 ∣ ℓ. This yields closed formulas for the splitting-field degree and an explicit finite-coordinate realization of Gal(Pℓ/ℚ), including its multiplication law, action on all 2ℓ roots, normal cyclic Kummer subgroup, multiplier character, center, stabilizers, commutator subgroup, and maximal abelian quotient. The paper proves that the Galois extension splits over its cyclotomic-quadratic quotient exactly when ℓ is odd and proves nonsplitting for every even ℓ. The derived subgroup is shown to be precisely the cyclic Kummer subgroup Cℓ, so the Galois groups are metabelian. The complete dihedral boundary is also determined: the Galois group is dihedral only for ℓ = 2 and ℓ = 3, giving respectively Gal(P₂/ℚ) ≅ D₈ and Gal(P₃/ℚ) ≅ D₁₂. The polynomial discriminant is computed exactly as Disc(Pℓ) = (−1)^(ℓ+1) ℓ^(2ℓ) 5^ℓ, and the finite ramification locus of the splitting field is proved to be exactly the set of primes p satisfying p ∣ 5ℓ. The analysis includes the special characteristic-5 degeneration and the corresponding conductor-5 field intersection. The resulting family provides an explicit all-exponent model in which quadratic arithmetic, cyclotomic intersections, Kummer theory, affine Galois actions, splitting obstructions, discriminants, and ramification can be described simultaneously and in closed form. (ARCHIVED PRIOR WORKING DESCRIPTION) This paper studies the Galois-theoretic properties of the polynomial family P_ℓ(x) = x^{2ℓ} - x^ℓ - 1 over Q, for integers ℓ ≥ 2 coprime to 5. We prove that the splitting field is Q(φ^{1/ℓ}, ζ_ℓ), where φ = (1 + √5)/2 is the golden ratio, and the field degree is [Q(φ^{1/ℓ}, ζ_ℓ):Q] = 2ℓ·φ(ℓ). The Galois group has order 2ℓ·φ(ℓ), acts transitively on the 2ℓ roots, and ensures irreducibility of P_ℓ(x) over Q. Computational verification with SymPy confirms theoretical predictions and yields explicit discriminants for ℓ = 2, 3, 4. The results unify Kummer and cyclotomic extensions, revealing a connection between the golden ratio and powered continued fractions.