  Suppose $m$ be a $12$-th power free integer. Let $K=\mathbb{Q}(\theta)$ be an
algebraic number field defined by a complex root $\theta$ of an irreducible
polynomial $x^{12}-m$ and $O_K$ be its ring of integers. In this paper, we
determine the highest power of $p$ dividing the index of the subgroup
$\Z[\theta]$ in $O_K$ and $p$-integral basis of $K$ for each prime $p$. These
$p$-integral bases lead to the construction of an integral basis of $K$ which
is illustrated with examples. In particular, when $m$ is a square free integer,
we provide necessary and sufficient conditions for the set
$\{1,\theta,\theta^2,\cdots,\theta^{10},\theta^{11}\}$ to be an integral basis
of $K.$
