Integral Basis for quartic Kummer extensions over $\mathbb{Z}[\iota]$

Abstract

Let $K=\mathbb{Q}[\iota]$ and $N=K[\sqrt[4]{\alpha}]$, $\alpha\in\mathbb{Z}[\iota]$, $alpha=fg^2h^3$, $f$, $g$, $h\in \mathbb{Z}[\iota]$ are pairwise coprime and square free. Let $\mathcal{O}_N$ be the ring of integers of $N$. In this article we construct normalised integral basis for $\mathcal{O}_N$ over $\mathbb{Z}[\iota]$, that is an integral basis of the form \[ \left\{1,\frac{f_1(\alpha)}{d_1},\frac{f_2(\alpha)}{d_2},\frac{f_{3}(\alpha)}{d_3}\right\} \] where $d_i \in \mathbb{Z}[i]$ and $f_i(X)$, $\leq i\leq 3$ are monic polynomials of degree $i$ over $\mathbb{Z}[\iota]$. We explicitly determine what $d_i$, $1\leq i\leq n-1$ are in terms of $f$, $g$ and $h$.