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

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 $|c|\leq200$.