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), 312323
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/370158 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
(47 #6648)
 [2]
Pierre
Barrucand, Sur certaines séries de Dirichlet, C. R.
Acad. Sci. Paris Sér. AB 269 (1969),
A294–A296 (French). MR 0246832
(40 #101)
 [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 (40 #2643)
 [4]
B.
D. Beach, H.
C. Williams, and C.
R. Zarnke, Some computer results on units in quadratic and cubic
fields, Proceedings of the TwentyFifth Summer Meeting of the Canadian
Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971) Lakehead
Univ., Thunder Bay, Ont., 1971, pp. 609–648. MR 0337887
(49 #2656)
 [5]
J.
W. S. Cassels, The rational solutions of the diophantine equation
𝑌²=𝑋³𝐷, Acta Math.
82 (1950), 243–273. MR 0035782
(12,11a)
 [6]
Harvey
Cohn, A numerical study of Dedekind’s cubic class number
formula, J. Res. Nat. Bur. Standards 59 (1957),
265–271. MR 0091308
(19,944a)
 [7]
R. DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern," J. Reine Angew. Math., v. 121, 1900, pp. 40123.
 [8]
Frank
Gerth III, Ranks of Sylow 3subgroups of ideal
class groups of certain cubic fields, Bull.
Amer. Math. Soc. 79
(1973), 521–525. MR 0314797
(47 #3347), http://dx.doi.org/10.1090/S000299041973131830
 [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 (Unpublished.)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
(45 #1877)
 [11]
Shinju
Kobayashi, On the 3rank of the ideal class groups of certain pure
cubic fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math.
20 (1973), 209–216. MR 0325572
(48 #3919)
 [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. 137.
 [13]
Carol
Neild and Daniel
Shanks, On the 3rank of quadratic fields and
the Euler product, Math. Comp. 28 (1974), 279–291. MR 0352042
(50 #4530), http://dx.doi.org/10.1090/S00255718197403520425
 [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
(18,286e)
 [15]
Daniel
Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 0352049
(50 #4537), http://dx.doi.org/10.1090/S00255718197403520498
 [1]
 I. O. ANGELL, "A table of complex cubic fields," Bull. London Math. Soc., v. 5, 1973, pp. 3738. MR 47 #6648. MR 0318099 (47:6648)
 [2]
 PIERRE BARRUCAND, "Sur certaines séries de Dirichlet," C. R. Acad. Sci. Paris Sér. AB, v. 269, 1969, pp. A294A296. MR 40 #101. MR 0246832 (40:101)
 [3]
 P. BARRUCAND & H. COHN, "A rational genus, class number divisibility, and unit theory for pure cubic fields," J. Number Theory, v. 2, 1970, pp. 721. MR 40 #2643. MR 0249398 (40:2643)
 [4]
 B. D. BEACH, H. C. WILLIAMS & C. R. ZARNKE, "Some computer results on units in quadratic and cubic fields," Proc. TwentyFifth Summer Meeting of the Canadian Math. Congress (Lakehead Univ., Thunder Bay, Ont., 1971), Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609648. MR 49 #2656. MR 0337887 (49:2656)
 [5]
 J. W. S. CASSELS, "The rational solutions of the diophantine equation ," Acta Math., v. 82, 1950, pp. 243273. MR 12, 11. MR 0035782 (12:11a)
 [6]
 HARVEY COHN, "A numerical study of Dedekind's cubic class number formula," J. Res. Nat. Bur. Standards, v. 59, 1957, pp. 265271. MR 19, 944. MR 0091308 (19:944a)
 [7]
 R. DEDEKIND, "Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern," J. Reine Angew. Math., v. 121, 1900, pp. 40123.
 [8]
 FRANK GERTH III, "Ranks of Sylow 3 subgroups of ideal class groups of certain cubic fields," Bull. Amer. Math. Soc., v. 79, 1973, pp. 521525. MR 0314797 (47:3347)
 [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, v. 3, 1971, pp. 712. MR 45 #1877. MR 0292795 (45:1877)
 [11]
 S. KOBAYASHI, "On the 3rank of the ideal class groups of certain pure cubic fields," J. Fac. Sci. Univ. Tokyo Sect. I A Math., v. 20, 1973, pp. 209216. MR 48 #3919. MR 0325572 (48:3919)
 [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. 137.
 [13]
 CAROL NEILD & DANIEL SHANKS, "On the 3rank of quadratic fields and the Euler product," Math. Comp., v. 28, 1974, pp. 279291. MR 0352042 (50:4530)
 [14]
 ERNST S. SELMER, "Tables for the purely cubic field ," Avh. Norske Vid. Akad. Oslo I, v. 1955, no. 5. MR 18, 286. MR 0080702 (18:286e)
 [15]
 DANIEL SHANKS, "The simplest cubic fields," Math. Comp., v. 28, 1974, pp. 11371152. MR 0352049 (50:4537)
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DOI:
http://dx.doi.org/10.1090/S00255718197603929139
PII:
S 00255718(1976)03929139
Article copyright:
© Copyright 1976
American Mathematical Society
