Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

TL;DR

This paper investigates the structure of power integral bases in certain quartic extensions of imaginary quadratic fields, reducing the problem to solving specific Pellian equations and explicitly characterizing all generators.

Contribution

It introduces a method to determine generators of relative power integral bases in a family of octic fields by solving associated Pellian equations.

Findings

Complete solution to the Pellian system for given parameters.

Explicit description of all generators of power integral bases.

Identification of conditions on parameter c for the generators.

Abstract

Let $M$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$ and let $\xi$ be a root of polynomial $$f\left( x\right) =x^{4}-2cx^{3}+2x^{2}+2cx+1,$$ where $c\in\mathbb{Z}_{M},$ $c\notin\left\{ 0,\pm2\right\}$. We consider an infinite family of octic fields $K_{c}=M\left( \xi\right)$ with the ring of integers $\mathbb{Z}_{K_{c}}.$ Our goal is to determine all generators of relative power integral basis of $\mathcal{O=}\mathbb{Z}_{M}\left[ \xi\right]$ over $\mathbb{Z}_{M}.$ We show that our problem reduces to solving the system of relative Pellian equations \[ cV^{2}-\left( c+2\right) U^{2}=-2\mu,\ \ cZ^{2}-\left( c-2\right) U^{2}=2\mu, \] where $\mu$ is an unit in $\mathbb{Z}_{M}$. We solve the system completely and find that all non-equivalent generators of power integral basis of $\mathcal{O}$ over $\mathbb{Z}_{M}$ are given by $\alpha=\xi,$ $2\xi-2c\xi ^{2}+\xi^{3}$ for $\left\vert c\right\vert \geq159108$ and $|c|\leq200$.