Integral Basis for quartic Kummer extensions over $\mathbb{Z}[{\hat{I}}^1]$

Abstract

Let $K=\mathbb{Q}[{\hat{I}}^1]$ and $N=K[\sqrt[4]{\hat{I}}\pm ]$, $\hat{I}\pm \in \mathbb{Z}[{\hat{I}}^1]$, $alpha=f{g}^2{h}^3$, $f$, $g$, $h\in \mathbb{Z}[{\hat{I}}^1]$ 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 $\mathbb{Z}[{\hat{I}}^1]$, that is an integral basis of the form \[ \left\{1,\frac{f_1(α)}{d_1},\frac{f_2(α)}{d_2},\frac{f_{3}(α)}{d_3}\right\} \] where $d_i\in \mathbb{Z}[i]$ and $f_i(X)$, $\le i\le 3$ are monic polynomials of degree $i$ over $\mathbb{Z}[{\hat{I}}^1]$. We explicitly determine what $d_i$, $1\le i\le n-1$ are in terms of $f$, $g$ and $h$.

29 pages