Integral basis of pure number fields of degree 16

Résumé

Let $\mathbb{L}=\mathbb{Q}(\theta)$ be a pure number field generated by a complex root $\theta$ of a monic irreducible polynomial $f(x)=x^{2^k}-d$, where $d\neq 1$ is a square free integer. In this work, we explicitly give an integral basis and the discriminant of $\mathbb{L}$ with $k=4$ by using an integral basis of the octic field $\mathbb{K}=\mathbb{Q}(\sqrt[8]d)$ considering the quadratic extension $\mathbb{L}\subset \mathbb{K}$. Furthermore, using the same approach, we present an integral basis and the discriminant of the pure field of degree $32.$