Monogenity of Fibonacci polynomials and Lucas polynomials

Abstract: We investigate the monogenity of irreducible factors of the Fibonacci polynomials $F_n(x)$ and the Lucas polynomials $L_n(x)$. Our main results show that for every odd positive integer $n$, all irreducible factors of $F_n(x)$ are monogenic, and for every even positive integer $n$, all irreducible factors of $L_n(x)$ are monogenic.

• The paper demonstrates that for odd n in Fibonacci polynomials and even n in Lucas polynomials, all irreducible factors yield monogenic extensions.
• It employs explicit root analysis and cyclotomic field theory to verify monogenity via discriminant computations and integral basis validations.
• The study provides practical methods for constructing monogenic extensions and outlines several open problems for further theoretical exploration.

Monogenity in Sequences of Fibonacci and Lucas Polynomials

Introduction and Problem Statement

This paper establishes definitive results on the monogenity of irreducible factors of Fibonacci polynomials $F_n(x)$ and Lucas polynomials $L_n(x)$. The notion of monogenity concerns the arithmetic structure of number fields $K$, which are said to be monogenic if their ring of integers admits a power integral basis, i.e., $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}_K$. Of particular interest is when irreducible polynomials $f(x) \in \mathbb{Z}[x]$ are themselves monogenic: that is, the field extension $K = \mathbb{Q}(\alpha)$ for a root $\alpha$ of $f$ satisfies $\mathcal{O}_K = \mathbb{Z}[\alpha]$.

While monogenity has been studied for static polynomial families, little is known about sequences defined by linear recurrences, such as $L_n(x)$0 and $L_n(x)$1. This paper systematically characterizes when irreducible factors of these polynomials yield monogenic extensions, discovering strong dependence on the parity of $L_n(x)$2. The authors prove the following major results:

• Theorem 1.1: For every odd $L_n(x)$3, all irreducible factors of $L_n(x)$4 are monogenic.
• Theorem 1.2: For every even $L_n(x)$5, all irreducible factors of $L_n(x)$6 are monogenic.

They further demonstrate that these parity restrictions are optimal and cannot be relaxed.

Technical Framework and Methodology

Polynomial Recurrences and Root Structure

The authors systematically leverage explicit descriptions of the roots of $L_n(x)$7 and $L_n(x)$8. Both polynomial sequences obey the characteristic recurrence $L_n(x)$9, with initial conditions distinguishing Fibonacci from Lucas polynomials.

Key properties:

• Fibonacci polynomials: $K$0 has roots $K$1 for $K$2, and explicit factorization into irreducibles follows from the decomposition of these roots across conjugacy classes determined by the Galois theory of cyclotomic fields.
• Lucas polynomials: $K$3 has roots $K$4 (range and indexation vary by $K$5's parity). The factor structure again maps onto certain subfields of $K$6.

The explicit correspondence between roots of these polynomials and traces or sums of roots of unity (real and imaginary parts) enables the monogenity analysis.

Galois Theory and Cyclotomic Subfields

A main technical engine is the algebraic analysis of field extensions of cyclotomic fields, specifically subfields generated by traces and sums of roots of unity. Results on discriminants and integral bases in these fields are employed, for instance, to address when rings like $K$7 or $K$8 constitute the full ring of integers.

Key results established include:

• When $K$9 and $\mathcal{O}_K = \mathbb{Z}[\theta]$0, $\mathcal{O}_K = \mathbb{Z}[\theta]$1, and the order $\mathcal{O}_K = \mathbb{Z}[\theta]$2 is maximal.
• For odd $\mathcal{O}_K = \mathbb{Z}[\theta]$3, the order $\mathcal{O}_K = \mathbb{Z}[\theta]$4 is maximal in $\mathcal{O}_K = \mathbb{Z}[\theta]$5.

These allow construction of minimal polynomials for the irreducible factors of $\mathcal{O}_K = \mathbb{Z}[\theta]$6 (for odd $\mathcal{O}_K = \mathbb{Z}[\theta]$7) and $\mathcal{O}_K = \mathbb{Z}[\theta]$8 (for even $\mathcal{O}_K = \mathbb{Z}[\theta]$9) whose roots generate monogenic fields.

Main Proofs and Parity Phenomena

The proof strategy for both theorems is uniform:

• Decompose the root set for $\theta \in \mathcal{O}_K$0 or $\theta \in \mathcal{O}_K$1 into Galois-stable classes corresponding to irreducible factors.
• Show that each minimal polynomial coincides (up to normalization) with the minimal polynomial of an explicit generator in a cyclotomic subfield, for which the authors verify monogenity rigorously using discriminant calculations and Galois-theoretic arguments.

A critical observation is that the structure of the root sets and corresponding field extensions is controlled by the parity of $\theta \in \mathcal{O}_K$2. Counterexamples for the converse cases (even $\theta \in \mathcal{O}_K$3 for $\theta \in \mathcal{O}_K$4, odd $\theta \in \mathcal{O}_K$5 for $\theta \in \mathcal{O}_K$6) demonstrate that monogenity fails for certain factors, and that these phenomena are intrinsic to the algebraic structure of the recurrences.

Numerical and Density Considerations

The authors supplement theoretical results with numerical experiments up to $\theta \in \mathcal{O}_K$7, indicating the rarity of monogenic irreducible factors outside the parity-constrained situations established in their theorems. The observation that there is no naive natural density for non-monogenic factors in the converse cases highlights open theoretical questions about the distribution and algebraic nature of such factors.

Implications and Open Problems

The characterization of monogenic irreducible factors for these polynomial sequences settles a natural class of questions in algebraic number theory regarding the arithmetic properties of polynomial recurrences and their factor fields. The findings have several ramifications:

• Practical: The result provides a method for constructing large families of monogenic extensions via recursively defined polynomial families, useful in computational algebraic number theory and related algorithmic problems.
• Theoretical: The work points to deep connections between the Galois-theoretic properties of recurrence sequences and the arithmetic of their factor fields, suggesting avenues for further study of "dynamic" polynomial families.

Key open problems posed include:

• Determining necessary and sufficient conditions for monogenity of all irreducible factors of $\theta \in \mathcal{O}_K$8 for even $\theta \in \mathcal{O}_K$9, and $f(x) \in \mathbb{Z}[x]$0 for odd $f(x) \in \mathbb{Z}[x]$1.
• Algorithmic characterization: whether the monogenity of all irreducible factors in these families can be decided directly from $f(x) \in \mathbb{Z}[x]$2's prime factorization and parity.

The investigation thus opens avenues for both a finer analytic characterization and computational techniques for monogenity in recurrence-based polynomial families.

Conclusion

This paper provides comprehensive results on the monogenity of irreducible factors of Fibonacci and Lucas polynomials, precisely delineated by the parity of the underlying index. The analysis relies on explicit root formulas, discriminant and integral basis computations in cyclotomic subfields, and Galois-theoretic methods, establishing that such dynamic families frequently yield monogenic extensions, but with sharp boundaries dictated by fundamental arithmetic properties. The systemic approach and open questions outlined are poised to inform further developments in computational and theoretical aspects of algebraic number theory.