On power basis of a class of number fields

Abstract

Let $f(x)=x^n+a{x}^2+bx+c\in \mathbb{Z}[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(θ)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let ${\mathbb{Z}}_K$ deonote the ring of algebraic integers of $K$. The aim of this paper is to provide the necessary and sufficient conditions involving only $a,c$ and $n$ for a given prime $p$ to divide the index of the subgroup $\mathbb{Z}[\theta ]$ in ${\mathbb{Z}}_K$. As a consequence, we provide families of monogenic algebraic number fields. Further, when ${\mathbb{Z}}_K\ne \mathbb{Z}[\theta ]$, we determine explicitly the index $[{\mathbb{Z}}_K:\mathbb{Z}[\theta ]]$ in some cases.