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

Authors:
Pierre Barrucand, H. C. Williams and L. Baniuk

Journal:
Math. Comp. **30** (1976), 312-323

MSC:
Primary 12A50; Secondary 12A30, 12A70

MathSciNet review:
0392913

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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 for 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.

**[1]**I. O. Angell,*A table of complex cubic fields*, Bull. London Math. Soc.**5**(1973), 37–38. MR**0318099****[2]**Pierre Barrucand,*Sur certaines séries de Dirichlet*, C. R. Acad. Sci. Paris Sér. A-B**269**(1969), A294–A296 (French). MR**0246832****[3]**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****[4]**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****[5]**J. W. S. Cassels,*The rational solutions of the diophantine equation 𝑌²=𝑋³-𝐷*, Acta Math.**82**(1950), 243–273. MR**0035782****[6]**Harvey Cohn,*A numerical study of Dedekind’s cubic class number formula*, J. Res. Nat. Bur. Standards**59**(1957), 265–271. MR**0091308****[7]**R. DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern,"*J. Reine Angew. Math.*, v. 121, 1900, pp. 40-123.**[8]**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**0314797**, 10.1090/S0002-9904-1973-13183-0**[9]**FRANK GERTH III, "The class structure of the pure cubic field." (Unpublished.)**[10]**Taira Honda,*Pure cubic fields whose class numbers are multiples of three*, J. Number Theory**3**(1971), 7–12. MR**0292795****[11]**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**0325572****[12]**A. MARKOV, "Sur les nombres entiers dépendants d'une racine cubique d'un nombre entier ordinaire,"*Mem. Acad. Imp. Sci. St. Petersburg*, v. (7) 38, 1892, no. 9, pp. 1-37.**[13]**Carol Neild and Daniel Shanks,*On the 3-rank of quadratic fields and the Euler product*, Math. Comp.**28**(1974), 279–291. MR**0352042**, 10.1090/S0025-5718-1974-0352042-5**[14]**Ernst S. Selmer,*Tables for the purely cubic field 𝐾(\root3\of𝑚)*, Avh. Norske Vid. Akad. Oslo. I.**1955**(1955), no. 5, 38. MR**0080702****[15]**Daniel Shanks,*The simplest cubic fields*, Math. Comp.**28**(1974), 1137–1152. MR**0352049**, 10.1090/S0025-5718-1974-0352049-8

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0392913-9

Article copyright:
© Copyright 1976
American Mathematical Society