What does this research mean for the field?

For every integer ℓ ≥ 2, the powered golden-ratio polynomial family Pℓ(x) = x^(2ℓ) − x^ℓ − 1 is irreducible over ℚ, with its splitting field and Galois group explicitly determined by cyclotomic and Kummer extensions of ℚ(√5). Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine the irreducibility, Galois groups, splitting field degrees, and arithmetic invariants of the powered golden-ratio polynomial family Pℓ(x) = x^(2ℓ) − x^ℓ − 1 for all integers ℓ ≥ 2.

August 19, 2026Open Access

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

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Implication

Theoretical analysis characterizes Galois groups and splitting fields for powered golden ratio polynomials, revealing metabelian structures and exact ramification loci across all exponents.

Key Points

To determine the irreducibility, Galois groups, splitting field degrees, and arithmetic invariants of the powered golden-ratio polynomial family Pℓ(x) = x^(2ℓ) − x^ℓ − 1 for all integers ℓ ≥ 2.
Analyzed splitting fields over the rational numbers using cyclotomic and Kummer extensions of ℚ(√5) governed by ℓ-th roots of the golden ratio.
Derived algebraic representations for group actions, subgroup lattices, splitting obstructions, and discriminants, supplemented with computational verification.
Proved Pℓ(x) is irreducible over ℚ for every ℓ ≥ 2, with the Galois extension splitting over its cyclotomic-quadratic quotient if and only if ℓ is odd.
Established that Gal(Pℓ/ℚ) is metabelian with derived subgroup Cℓ, matching dihedral groups only at ℓ = 2 (D₈) and ℓ = 3 (D₁₂).
Calculated the exact discriminant Disc(Pℓ) = (−1)^(ℓ+1) ℓ^(2ℓ) 5^ℓ and proved the finite ramification locus consists precisely of primes dividing 5ℓ.

Cite This Study

A 2026 study studied this question.

synapsesocial.com/papers/6a8562eb03308d306e2d5e10 https://doi.org/10.5281/zenodo.17635736

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Implication

Theoretical analysis characterizes Galois groups and splitting fields for powered golden ratio polynomials, revealing metabelian structures and exact ramification loci across all exponents.

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

Discussion

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Implication

Key Points

Cite This Study

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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