On the monogenicity of power-compositional Shanks polynomials

Abstract

Let $f(x)\in \mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $\mathbb{Q}$. We say $f(x)$ is \emph{monogenic} if $\Theta=\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers $\mathbb{Z}_K$ of $K=\mathbb{Q}(\theta)$, where $f(\theta)=0$. If $\Theta$ is not a basis for $\mathbb{Z}_K$, we say that $f(x)$ is \emph{non-monogenic}.Let $k\ge 1$ be an integer, and let $(U_n)$be the sequence defined by \[U_0=U_1=0,\qquad U_2=1 \qquad \text{and}\qquad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} \qquad \text{for $n\ge 3$}.\] It is well known that $(U_n)$ is periodic modulo any integer $m\ge 2$, and we let $\pi(m)$ denote the length of this period. We define a \emph{$k$-Shanks prime} to be a prime $p$ such that $\pi(p^2)=\pi(p)$. Let $\mathcal{S}_k(x)=x^{3}-kx^{2}-(k+3)x-1$ and $\mathcal{D}=(k^2+3k+9)/\gcd(3,k)^2$. Suppose that $k\not \equiv 3 \pmod{9}$ and that $\mathcal{D}$ is squarefree. In this article, we prove that $p$ is a $k$-Shanks prime if and only if $\mathcal{S}_k(x^p)$ is non-monogenic, for any prime $p$ such that $\mathcal{S}_k(x)$ is irreducible in $\mathbb{F}_p[x]$. Furthermore, we show that $\mathcal{S}_k(x^p)$ is monogenic for any prime divisor $p$ of $\mathcal{D}$. These results extend previous work of the author on $k$-Wall-Sun-Sun primes.