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

TL;DR

This paper constructs explicit normalized integral bases for quartic Kummer extensions over the Gaussian integers, providing formulas for the basis elements in terms of the defining parameters.

Contribution

It introduces a method to explicitly determine the integral basis of quartic Kummer extensions over [ ext{iota}] in terms of the parameters defining the extension.

Findings

Explicit formulas for the basis denominators $d_i$ in terms of $f$, $g$, and $h$

Construction of a normalized integral basis for the extension

Explicit description of basis elements involving polynomials $f_i(\alpha)$

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 $h$.