Improving the speed of calculating the regulator of certain pure cubic fields

Author:
H. C. Williams

Journal:
Math. Comp. **35** (1980), 1423-1434

MSC:
Primary 12A30

DOI:
https://doi.org/10.1090/S0025-5718-1980-0583520-4

MathSciNet review:
583520

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Abstract: To calculate *R*, the regulator of a pure cubic field $Q(\sqrt [3]{{D)}}$, a complete period of Voronoiβs continued fraction algorithm over $Q(\sqrt [3]{{D)}}$ is usually generated. In this paper it is shown how, in certain pure cubic fields, *R* can be determined by generating only about one third of this period. These results were used on a computer to find *R* and then the class number for all pure cubic fields $Q(\sqrt [3]{{p)}}$, where *p* is a prime, $p \equiv - 1\;\pmod 3$, and $p < 2 \times {10^5}$. Graphs illustrating the distribution of such cubic fields with class number one are presented.

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*On a Generalization of the Algorithm of Continued Fractions*, Doctoral Dissertation, Warsaw, 1896. (Russian)

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Article copyright:
© Copyright 1980
American Mathematical Society