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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Improving the speed of calculating the regulator of certain pure cubic fields
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by H. C. Williams PDF
Math. Comp. 35 (1980), 1423-1434 Request permission

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.
References
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Math. Comp. 35 (1980), 1423-1434
  • MSC: Primary 12A30
  • DOI: https://doi.org/10.1090/S0025-5718-1980-0583520-4
  • MathSciNet review: 583520