Monogenic Cyclic Polynomials in Recurrence Sequences
Joshua Harrington, Lenny Jones
TL;DR
The paper constructs a polynomial recurrence sequence $\{w_n(x)\}$ with $w_0=1$, $w_1=1$, and $w_n=(x-2)w_{n-1}-w_{n-2}$, and proves that for $n\ge 2$ each $w_n(x)$ factors into $\Omega_d(x)$ over divisors $d>1$ of $2n-1$, where each $\Omega_d$ is monogenic of degree $\phi(d)/2$; moreover, $\Omega_d$ is cyclic exactly when $d$ satisfies Condition ${\mathcal C}$. The core method expresses $\Omega_d(x)$ as a product over carefully chosen real roots, shows its splitting field is the maximal real subfield of a cyclotomic field, and applies Washington’s results to obtain monogenicity, with cyclicity governed by ${\mathcal C}$. A key corollary shows that if $2n-1$ satisfies ${\mathcal C}$, all irreducible factors of $w_n(x)$ are monogenic and cyclic, and distinct $n$-values yield non-equivalent factors. The Final Comments connect primitive divisors $\Omega_{2n-1}(x)$ to maximal real subfields $K_N^+$, establishing a link between the recurrence construction and cyclotomic real subfields, and implying the existence of infinitely many monogenic cyclic polynomials arising from these subfields.
Abstract
Let $f(x)\in {\mathbb Z}[x]$ be an $N$th degree polynomial that is monic and irreducible over ${\mathbb Q}$. We say that $f(x)$ is {\em monogenic} if $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. We say that $f(x)$ is {\em cyclic} if the Galois group of $f(x)$ over ${\mathbb Q}$ is the cyclic group of order $N$. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences.