Computing power integral bases in quartic relative extensions

[摘要] We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (I. Gaal, A. Petho, and M. Pohst, 1993, J. Symbolic Comput. 16, 563-584, 1996, J. Number Theory 57, 90-104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations determine the generators of power integral bases of K over M up to translation by integers of M and multiplication by unit factors of M. The new method is based on our ability to solve relative Thue equations efficiently by the algorithm in (I. Gaal and M. Pohst, 2000, Math. Comp., to appear). In the case K is an octic field with a quadratic subfield M we can also consider the absolute index of elements of K, having relative index 1 over M. In order to determine all generators of power integral bases of K (over Q) we determine the corresponding translating elements and unit factors properly. This is done by solving. an equation similar to an inhomogeneous Thue equation. We illustrate our algorithms with detailed examples. (C) 2000 Academic Press.

[发布日期] 2000-12-01

[关键词] power integral bases;resolution of index form equations;octic fields