Monogenic Cyclic Polynomials in Recurrence Sequences

TL;DR

This paper investigates the occurrence of monogenic cyclic polynomials within specific polynomial recurrence sequences, focusing on their algebraic properties and Galois groups.

Contribution

It provides new insights into the conditions under which monogenic cyclic polynomials appear in polynomial recurrence sequences.

Findings

Identification of criteria for monogenic cyclic polynomials in recurrence sequences

Characterization of Galois groups associated with these polynomials

Examples illustrating the occurrence of such polynomials

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