DISCRIMINANT and integral basis OF $\mathbb{Q}(\sqrt[12]{a})$

Abstract

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 $\mathbb{Z}[\theta ]$ 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 $K.$