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

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701638-2

MathSciNet review:
701638

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Abstract: Let be the algebraic number field formed by adjoining to the rationals . Let *R* and *h* be, respectively, the regulator and class number of . Shanks has described a method of evaluating *R* for , 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 *n*th to the *m*th step, where *m* is approximately 2*n*, in the continued fraction expansion of . 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 , *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 operations. When *h* is small, this is an improvement over the operations required by Voronoi's algorithm to find *R*. For example, with , it required only 5 minutes for an AMDAHL 470/*V*7 computer to find that and . This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm.

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0701638-2

Article copyright:
© Copyright 1983
American Mathematical Society