Non-monogenic factors of the Fibonacci and Lucas polynomials: the order ℤ[ζₙ-ζₙ⁻¹] for odd n

Abstract

Let Fₙ and Lₙ be the Fibonacci and Lucas polynomials, defined by F₀=0, F₁=1, L₀=2, L₁=x and the common recurrence Pₖ₊₂=xPₖ₊₁+Pₖ. A recent preprint of Chen, Guo and Hong proves that every irreducible factor of Fₙ is monogenic when n is odd and that every irreducible factor of Lₙ is monogenic when n is even, observes that neither parity hypothesis can be dropped, and asks, as its Problems 1–3, for necessary and sufficient conditions in the two remaining parities. We settle the negative half uniformly in n. For every odd n>1, the element β=ζₙ-ζₙ⁻¹ generates the whole cyclotomic field ℚ(ζₙ) and its minimal polynomial divides Lₙ, yet ℤ[β] is a proper subring of the ring of integers of ℚ(ζₙ): one has 2ζₙ ∈ ℤ[β] but ζₙ ∉ ℤ[β]. Consequently Lₙ has a non-monogenic irreducible factor for every odd n>1, and Fₘ has one for every even m that is not a power of 2; in particular, for odd m every irreducible factor of Lₘ is monogenic if and only if m=1, which answers the second of the three problems outright. The proof uses no discriminant, no Dedekind criterion and no Newton polygon. The Binet identity Lₖ(u-v)=uᵏ+(-1)ᵏvᵏ, valid whenever uv=1, puts β on Lₙ; the conjugation ζ ↦ ζ⁻¹ sends β to -β, so ζₙ ∈ ℤ[β] would make β twice an algebraic integer, and a geometric sum then makes n twice an algebraic integer as well. Individual cases were known — n=3 classically, and the degrees corresponding to n=5,7,9,11,15,23 in work of Jones, Harrington–Jones and König — but we have not found the statement uniform in n, nor this mechanism, in the literature. Combining the theorem with the two positive propositions of the source classifies the monogenic factors of Fₙ and Lₙ completely and answers all three problems; that combination, the exact index 2^(φ(n)/2) and a confirming computation are recorded as remarks, outside the formal development and labelled as such. Every lemma, proposition, theorem and corollary of Sections [sec:prelim]–[sec:anchors] is machine-checked in Lean 4 against Mathlib.

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