On primes dividing the index of a quadrinomial

Abstract

Let ${\mathbb{Z}}_K$ denote the ring of algebraic integers of an algebraic number field $K=\mathbb{Q}\left(𝜃\right)$ where the algebraic integer $𝜃$ is a root of an irreducible quadrinomial $f\left(x\right)=x^n+a{x}^{n-1}+b{x}^{n-2}+c$ belonging to $\mathbb{Z}\left[x\right]$ with $a^2=4b$. We give necessary and sufficient conditions involving only $a,b,c,n$ for a prime $p$ to divide the index of the subgroup $\mathbb{Z}\left[𝜃\right]$ in ${\mathbb{Z}}_K$. As a consequence, we obtain necessary and sufficient conditions for ${\mathbb{Z}}_K$ to be equal to $\mathbb{Z}\left[𝜃\right]$. Moreover, when ${\mathbb{Z}}_K\ne \mathbb{Z}\left[𝜃\right]$, we provide an explicit formula for the index $\left[{\mathbb{Z}}_K:\mathbb{Z}\left[𝜃\right]\right]$ in some cases.