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Mathematics of Computation

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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
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 Voronoi’s 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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Article copyright: © Copyright 1980 American Mathematical Society