zbMATH — the first resource for mathematics

The computation of the fundamental unit of totally complex quartic orders. (English) Zbl 0627.12004
It is shown how the generalized Voronoi algorithm can be used to produce a practical algorithm for determining the regulator and/or fundamental unit of an arbitrary totally complex quartic order. The author shows that his algorithm will yield a fundamental unit of any such order in O(R \(D^{\epsilon})\) binary operations (for every \(\epsilon >0)\), where D is the absolute value of the discriminant and R is the regulator of the order.
An analogue of Galois’ theorem on the symmetry of the continued fraction expansion of the square root of a rational number is also established. The paper concludes with a table of computational results for the orders \({\mathbb{Z}}[^ 4\sqrt{-d}]\) with \(1\leq d\leq 500\).
Reviewer: H.C.Williams

11R16 Cubic and quartic extensions
11R27 Units and factorization
12-04 Software, source code, etc. for problems pertaining to field theory
Full Text: DOI