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

Let $K=\mathbb{Q}[\iota ]$ and $N=K[\sqrt[4]{\alpha }]$, $\alpha \in \mathbb{Z}[\iota ]$, $alpha=f{g}^2{h}^3$, $f$, $g$, $h\in \mathbb{Z}[\iota ]$ 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 $$\{1,\frac{f_1(\alpha )}{d_1},\frac{f_2(\alpha )}{d_2},\frac{f_3(\alpha )}{d_3}\}$$ where $d_i\in \mathbb{Z}[i]$ and $f_i(X)$, $\le i\le 3$ are monic polynomials of degree $i$ over $\mathbb{Z}[\iota ]$. We explicitly determine what $d_i$, $1\le i\le n-1$ are in terms of $f$, $g$ and $h$.