Improving the speed of calculating the regulator of certain pure cubic fields
Author:
H. C. Williams
Journal:
Math. Comp. 35 (1980), 14231434
MSC:
Primary 12A30
MathSciNet review:
583520
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Abstract: To calculate R, the regulator of a pure cubic field , a complete period of Voronoi's continued fraction algorithm over 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 , where p is a prime, , and . Graphs illustrating the distribution of such cubic fields with class number one are presented.
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 PIERRE BARRUCAND & HARVEY COHN, "A rational genus, class number divisibility, and unit theory for pure cubic fields," J. Number Theory, v. 2, 1970, pp. 721. MR 0249398 (40:2643)
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 [4]
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 TAIRA HONDA, "Pure cubic fields whose class numbers are multiples of three," J. Number Theory, v. 3, 1971, pp. 712. MR 0292795 (45:1877)
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 R. STEINER, "On the units in algebraic number fields," Proc. 6th Manitoba Conf. on Numerical Math., Winnipeg, 1976, pp. 413435. MR 532716 (81b:12008)
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 G. F. VORONOI, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
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 H. C. WILLIAMS & J. BROERE, "A computational technique for evaluating and the class number of a real quadratic field," Math. Comp., v. 30, 1976, pp. 887893. MR 0414522 (54:2623)
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 H. C. WILLIAMS, "Certain pure cubic fields with class number one," Math. Comp., v. 31, 1977, pp. 578580; "Corrigendum", Math. Comp., v. 33, 1979, pp. 847848. MR 0432591 (55:5578)
 [11]
 H. C. WILLIAMS & P. A. BUHR, "Calculation of the regulator of by use of the nearest integer continued fraction algorithm," Math. Comp., v. 33, 1979, pp. 364381. MR 514833 (80e:12003)
 [12]
 H. C. WILLIAMS & DANIEL SHANKS, "A note on class number one in pure cubic fields," Math. Comp., v. 33, 1979, pp. 13171320. MR 537977 (80g:12002)
 [13]
 H. C. WILLIAMS, G. CORMACK & E. SEAH, "Calculation of the regulator of a pure cubic field," Math. Comp., v. 34, 1980, pp. 567611. MR 559205 (81d:12003)
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DOI:
http://dx.doi.org/10.1090/S00255718198005835204
PII:
S 00255718(1980)05835204
Article copyright:
© Copyright 1980
American Mathematical Society
