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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

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