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