Discriminant and integral basis of pure nonic fields

Let $K=\text{Q}(\theta )$ be an algebraic number field with $\theta$ satisfying an irreducible polynomial $x^9-a$ over the field $\text{Q}$ of rationals and ${\mathbb{Z}}_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact power of each prime which divides the index of the subgroup $\mathbb{Z}[\theta ]$ in ${\mathbb{Z}}_K$. Further, we give a $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to a construction of an integral basis of $K$ which is illustrated with examples.