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

Williams, H. C.

doi:10.1090/S0025-5718-1980-0583520-4

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

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 249398, DOI 10.1016/0022-314X(70)90003-X
Pierre Barrucand and Harvey Cohn, Remarks on principal factors in a relative cubic field, J. Number Theory 3 (1971), 226–239. MR 276197, DOI 10.1016/0022-314X(71)90040-0
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 392913, DOI 10.1090/S0025-5718-1976-0392913-9
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
Taira Honda, Pure cubic fields whose class numbers are multiples of three, J. Number Theory 3 (1971), 7–12. MR 292795, DOI 10.1016/0022-314X(71)90045-X
Richard B. Lakein, Computation of the ideal class group of certain complex quartic fields. II, Math. Comp. 29 (1975), 137–144. MR 444605, DOI 10.1090/S0025-5718-1975-0444605-4
Ray Steiner, On the units in algebraic number fields, Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976) Congress. Numer., XVIII, Utilitas Math., Winnipeg, Man., 1977, pp. 413–435. MR 532716

On a Generalization of the Algorithm of Continued Fractions

H. C. Williams and J. Broere, A computational technique for evaluating $L(1,\chi )$ and the class number of a real quadratic field, Math. Comp. 30 (1976), no. 136, 887–893. MR 414522, DOI 10.1090/S0025-5718-1976-0414522-5
H. C. Williams, Certain pure cubic fields with class-number one, Math. Comp. 31 (1977), no. 138, 578–580. MR 432591, DOI 10.1090/S0025-5718-1977-0432591-4
H. C. Williams and P. A. Buhr, Calculation of the regulator of $\textbf {Q}(\surd D)$ by use of the nearest integer continued fraction algorithm, Math. Comp. 33 (1979), no. 145, 369–381. MR 514833, DOI 10.1090/S0025-5718-1979-0514833-1
H. C. Williams and Daniel Shanks, A note on class-number one in pure cubic fields, Math. Comp. 33 (1979), no. 148, 1317–1320. MR 537977, DOI 10.1090/S0025-5718-1979-0537977-7
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, DOI 10.1090/S0025-5718-1980-0559205-7