On relative power integral basis of a family of numbers fields

Abstract

Let $K$ be a number field with ring of integers $R$ and $L=K\left(\alpha \right)$ where $\alpha$ satisfies the irreducible polynomial $P\left(X\right)=X^{3^n}+a{X}^{3^s}-b$ of $R\left[X\right]$ $\left(n,s\in \mathbb{N},n>s\right)$. We give necessary and sufficient conditions that involve only $a$, $b$, $s$, $n$ to study the monogeneity of $L$ over $K$. We also present some applications, giving integral bases of some extensions of degree $2\cdot 3^n$, as well, we give their absolute discriminant.