Binomial Thue equations and power integral bases in pure quartic fields

Abstract

It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating power integral bases. In this paper we determine power integral bases in pure quartic fields by reducing the corresponding index form equation to quartic binomial Thue equations. We apply recent results on the solutions of binomial Thue equations of type $a^4-mb^4=\pm 1\; ({\rm in}\; a,b\in Z)$ as well as we perform an extensive calculation by a high performance computer to determine "small" solutions (with $\max(|a|,|b|)<10^{500}$) of binomial Thue equations for $m<10^7$. Using these results on binomial Thue equations we characterize power integral bases in infinite subfamilies of pure quartic fields. For $1< m< 10^7$ we give all generators of power integral bases with coefficients $<10^{1000}$ in the integral basis.