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.