On Power Basis of a Class of Number Fields

Abstract

Let $f(x)=x^n+ax^2+bx+c \in {\textbf {Z}}[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K={\textbf {Q}}(\theta )$ be an algebraic number field defined by a complex root $\theta$ of f(x). Let ${\textbf {Z}}_K$ denote 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 ${\textbf {Z}}[\theta ]$ in ${\textbf {Z}}_K$. As a consequence, we provide families of monogenic algebraic number fields. Further, when ${\textbf {Z}}_K \ne {\textbf {Z}}[\theta ]$, we determine explicitly the index $[{\textbf {Z}}_K: {\textbf {Z}}[\theta ]]$ in some cases.