Monogenic cyclic polynomials in recurrence sequences
Volume 180 / 2026
Colloquium Mathematicum 180 (2026), 21-30
MSC: Primary 11R09; Secondary 11B39, 11R18, 11R32
DOI: 10.4064/cm9671-10-2025
Published online: 28 January 2026
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 monogenic if $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of $\mathbb Q(\theta )$, where $f(\theta )=0$. We say that $f(x)$ is 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.
