Integral basis for quartic Kummer extensions over \(\mathbb {Q}[\iota ]\)

Let \(K=\mathbb {Q}[\iota ]\) $K=\mathbb{Q}[\iota ]$ and \(N=K[\root 4 \of {\alpha }]\) $N=K[\sqrt[4]{\alpha }]$ , \(\alpha \in \mathbb {Z}[\iota ]\) $\alpha \in \mathbb{Z}[\iota ]$ , \(\alpha =fg^2h^3\) $\alpha =f{g}^2{h}^3$ , \(f\) $f$ , \(g\) $g$ , \(h\in \mathbb {Z}[\iota ]\) $h\in \mathbb{Z}[\iota ]$ are pairwise coprime and square free. Let \(\mathcal{O}_{ \scriptstyle N}\) ${\mathcal{O}}_N$ be the ring of integers of \(N\) $N$ . In this article we construct normalised integral basis for \(\mathcal{O}_{ \scriptstyle N}\) ${\mathcal{O}}_N$ over \(\mathbb {Z}[\iota ]\) $\mathbb{Z}[\iota ]$ , that is an integral basis of the form \(\begin{aligned} \left\{ 1,\frac{f_1(\alpha )}{d_1},\frac{f_2(\alpha )}{d_2},\frac{f_{3}(\alpha )}{d_3}\right\} \end{aligned}\) $$\begin{array}{c}\hfill \left\{1,,,\frac{f_1(\alpha )}{d_1},,,\frac{f_2(\alpha )}{d_2},,,\frac{f_3(\alpha )}{d_3}\right\}\end{array}$$ where \(d_i\in \mathbb {Z}[\iota ]\) $d_i\in \mathbb{Z}[\iota ]$ and \(f_i(X)\) $f_i(X)$ , \(1\le i\le 3\) $1\le i\le 3$ are monic polynomials of degree \(i\) $i$ over \(\mathbb {Z}[\iota ]\) $\mathbb{Z}[\iota ]$ . We explicitly determine what \(d_i\) $d_i$ , \(1\le i\le 3\) $1\le i\le 3$ are in terms of \(f\) $f$ , \(g\) $g$ and \(h\) $h$ .