Monogenic Cyclic Polynomials in Recurrence Sequences

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 $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ is a basis for the ring of integers of $\mathbb{Q}(\theta )$, where $f(\theta )=0$. We say that $f(x)$ is {\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.