A computational technique for determining the class number of a pure cubic field

Barrucand, Pierre; Williams, H. C.; Baniuk, L.

doi:10.1090/S0025-5718-1976-0392913-9

A computational technique for determining the class number of a pure cubic field
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by Pierre Barrucand, H. C. Williams and L. Baniuk PDF

Math. Comp. 30 (1976), 312-323 Request permission

Abstract:

Two different computational techniques for determining the class number of a pure cubic field are discussed. These techniques were implemented on an IBM/370-158 computer, and the class number for each pure cubic field $Q({D^{1/3}})$ for $D = 2,3, \ldots ,9999$ was obtained. Several tables are presented which summarize the results of these computations. Some theorems concerning the class group structure of pure cubic fields are also given. The paper closes with some conjectures which were inspired by the computer results.

References

I. O. Angell, A table of complex cubic fields, Bull. London Math. Soc. 5 (1973), 37–38. MR 318099, DOI 10.1112/blms/5.1.37
Pierre Barrucand, Sur certaines séries de Dirichlet, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A294–A296 (French). MR 246832
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
B. D. Beach, H. C. Williams, and C. R. Zarnke, Some computer results on units in quadratic and cubic fields, Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971) Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609–648. MR 0337887
J. W. S. Cassels, The rational solutions of the diophantine equation $Y^2=X^3-D$, Acta Math. 82 (1950), 243–273. MR 35782, DOI 10.1007/BF02398279
Harvey Cohn, A numerical study of Dedekind’s cubic class number formula, J. Res. Nat. Bur. Standards 59 (1957), 265–271. MR 0091308, DOI 10.6028/jres.059.031

J. Reine Angew. Math.

Frank Gerth III, Ranks of Sylow $3$-subgroups of ideal class groups of certain cubic fields, Bull. Amer. Math. Soc. 79 (1973), 521–525. MR 314797, DOI 10.1090/S0002-9904-1973-13183-0
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
Shinju Kobayashi, On the $3$-rank of the ideal class groups of certain pure cubic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 209–216. MR 325572

Mem. Acad. Imp. Sci. St. Petersburg

Carol Neild and Daniel Shanks, On the $3$-rank of quadratic fields and the Euler product, Math. Comp. 28 (1974), 279–291. MR 352042, DOI 10.1090/S0025-5718-1974-0352042-5
Ernst S. Selmer, Tables for the purely cubic field $K(\root 3\of m)$, Avh. Norske Vid.-Akad. Oslo I 1955 (1955), no. 5, 38. MR 80702
Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8