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Mathematics of Computation

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A rapid method of evaluating the regulator and class number of a pure cubic field

Authors: H. C. Williams, G. W. Dueck and B. K. Schmid
Journal: Math. Comp. 41 (1983), 235-286
MSC: Primary 12A50; Secondary 12-04, 12A30, 12A45
MathSciNet review: 701638
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Abstract: Let $ \mathcal{K} = \mathcal{Q}(\theta )$ be the algebraic number field formed by adjoining $ \theta $ to the rationals $ \mathcal{Q}$. Let R and h be, respectively, the regulator and class number of $ \mathcal{K}$. Shanks has described a method of evaluating R for $ \mathcal{Q}(\sqrt D )$, where D is a positive integer. His technique improved the speed of the usual continued fraction algorithm for finding R by allowing one to proceed almost directly from the nth to the mth step, where m is approximately 2n, in the continued fraction expansion of $ \sqrt D $. This paper shows how Shanks' idea can be extended to the Voronoi algorithm, which is used to find R in cubic fields of negative discriminant. It also discusses at length an algorithm for finding R and h for pure cubic fields $ \mathcal{Q}(\sqrt[3]{D})$, D an integer. Under a certain generalized Riemann Hypothesis the ideas developed here will provide a new method which will find R and h in $ O({D^{2/5 + \varepsilon }})$ operations. When h is small, this is an improvement over the $ O(D/h)$ operations required by Voronoi's algorithm to find R. For example, with $ D = 200171999$, it required only 5 minutes for an AMDAHL 470/V7 computer to find that $ R = 518594546.969083$ and $ h = 1$. This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm.

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Article copyright: © Copyright 1983 American Mathematical Society