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.
 [1]
Pierre
Barrucand and Harvey
Cohn, A rational genus, class number divisibility, and unit theory
for pure cubic fields, J. Number Theory 2 (1970),
7–21. MR
0249398 (40 #2643)
 [2]
Pierre
Barrucand and Harvey
Cohn, Remarks on principal factors in a relative cubic field,
J. Number Theory 3 (1971), 226–239. MR 0276197
(43 #1945)
 [3]
Pierre
Barrucand, H.
C. Williams, and L.
Baniuk, A computational technique for
determining the class number of a pure cubic field, Math. Comp. 30 (1976), no. 134, 312–323. MR 0392913
(52 #13726), http://dx.doi.org/10.1090/S00255718197603929139
 [4]
B.
N. Delone and D.
K. Faddeev, The theory of irrationalities of the third degree,
Translations of Mathematical Monographs, Vol. 10, American Mathematical
Society, Providence, R.I., 1964. MR 0160744
(28 #3955)
 [5]
Taira
Honda, Pure cubic fields whose class numbers are multiples of
three, J. Number Theory 3 (1971), 7–12. MR 0292795
(45 #1877)
 [6]
Richard
B. Lakein, Computation of the ideal class group
of certain complex quartic fields. II, Math.
Comp. 29 (1975),
137–144. Collection of articles dedicated to Derrick Henry Lehmer on
the occasion of his seventieth birthday. MR 0444605
(56 #2955), http://dx.doi.org/10.1090/S00255718197504446054
 [7]
Ray
Steiner, On the units in algebraic number fields, (Univ.
Manitoba, Winnipeg, Man., 1976) Congress. Numer., XVIII, Utilitas Math.,
Winnipeg, Man., 1977, pp. 413–435. MR 532716
(81b:12008)
 [8]
G. F. VORONOI, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
 [9]
H.
C. Williams and J.
Broere, A computational technique for
evaluating 𝐿(1,𝜒) and the class number of a real quadratic
field, Math. Comp. 30
(1976), no. 136, 887–893. MR 0414522
(54 #2623), http://dx.doi.org/10.1090/S00255718197604145225
 [10]
H.
C. Williams, Certain pure cubic fields with
classnumber one, Math. Comp.
31 (1977), no. 138, 578–580. MR 0432591
(55 #5578), http://dx.doi.org/10.1090/S00255718197704325914
 [11]
H.
C. Williams and P.
A. Buhr, Calculation of the regulator of
𝑄(√𝐷) by use of the nearest integer continued
fraction algorithm, Math. Comp.
33 (1979), no. 145, 369–381. MR 514833
(80e:12003), http://dx.doi.org/10.1090/S00255718197905148331
 [12]
H.
C. Williams and Daniel
Shanks, A note on classnumber one in pure
cubic fields, Math. Comp.
33 (1979), no. 148, 1317–1320. MR 537977
(80g:12002), http://dx.doi.org/10.1090/S00255718197905379777
 [13]
H.
C. Williams, G.
Cormack, and E.
Seah, Calculation of the regulator of a pure
cubic field, Math. Comp.
34 (1980), no. 150, 567–611. MR 559205
(81d:12003), http://dx.doi.org/10.1090/S00255718198005592057
 [1]
 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)
 [2]
 PIERRE BARRUCAND & HARVEY COHN, "Remarks on principal factors in a relative cubic field," J. Number Theory, v. 3, 1971, pp. 226239. MR 0276197 (43:1945)
 [3]
 PIERRE BARRUCAND, H. C. WILLIAMS & L. BANIUK, "A computational technique for determining the class number of a pure cubic field," Math. Comp., v. 30, 1976, pp. 312323. MR 0392913 (52:13726)
 [4]
 B. N. DELONE & D. K. FADDEEV, The Theory of Irrationalities of the Third Degree, Transl. Math. Monographs, Vol. 10, Amer. Math. Soc., Providence, R.I., 1964. MR 0160744 (28:3955)
 [5]
 TAIRA HONDA, "Pure cubic fields whose class numbers are multiples of three," J. Number Theory, v. 3, 1971, pp. 712. MR 0292795 (45:1877)
 [6]
 R. B. LAKEIN, "Review of UMT File: Table of Class Numbers Greater than 1, for Fields , ," Math. Comp., v. 29, 1975, pp. 335336. MR 0444605 (56:2955)
 [7]
 R. STEINER, "On the units in algebraic number fields," Proc. 6th Manitoba Conf. on Numerical Math., Winnipeg, 1976, pp. 413435. MR 532716 (81b:12008)
 [8]
 G. F. VORONOI, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
 [9]
 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)
 [10]
 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
