Power integral bases in a family of sextic fields with quadratic subfields

Abstract

Let $M=\Q(i\sqrt{d})$ be any imaginary quadratic field with a positive
square-free $d$. Consider the polynomial
\[
f(x)=x^3-ax^2-(a+3)x-1,
\]
with a parameter $a\in\Z$.
Let $K=M(\alpha)$, where $\alpha$ is a root of $f$.
This is an infinite parametric family of sextic fields depending
on two parameters, $a$ and $d$.
Applying relative Thue equations
we determine the relative power integral bases of these sextic fields
over their quadratic subfields.
Using these results we also determine generators of (absolute)
power integral bases of the sextic fields.