The exponent three class group problem for some real cyclic cubic number fields
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- by Stéphane Louboutin
- Proc. Amer. Math. Soc. 130 (2002), 353-361
- DOI: https://doi.org/10.1090/S0002-9939-01-06168-8
- Published electronically: June 8, 2001
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Abstract:
We determine all the simplest cubic fields whose ideal class groups have exponent dividing $3$, thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number $1$ and the determination by D. Byeon of all all the simplest cubic fields with class number $3$. We prove that there are $23$ simplest cubic fields with ideal class groups of exponent $3$ (and $8$ simplest cubic fields with ideal class groups of exponent $1$, i.e. with class number one).References
- Eric Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), no. 191, 355–380. MR 1023756, DOI 10.1090/S0025-5718-1990-1023756-8
- Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717–1735. MR 1355006, DOI 10.1090/S0025-5718-96-00763-6
- Johannes Buchmann and H. C. Williams, On the computation of the class number of an algebraic number field, Math. Comp. 53 (1989), no. 188, 679–688. MR 979937, DOI 10.1090/S0025-5718-1989-0979937-4
- Dongho Byeon, Class number $3$ problem for the simplest cubic fields, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1319–1323. MR 1664337, DOI 10.1090/S0002-9939-99-05330-7
- T. W. Cusick, Lower bounds for regulators, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 63–73. MR 756083, DOI 10.1007/BFb0099441
- Günter Lettl, A lower bound for the class number of certain cubic number fields, Math. Comp. 46 (1986), no. 174, 659–666. MR 829636, DOI 10.1090/S0025-5718-1986-0829636-1
- Stéphane Louboutin, Lower bounds for relative class numbers of CM-fields, Proc. Amer. Math. Soc. 120 (1994), no. 2, 425–434. MR 1169041, DOI 10.1090/S0002-9939-1994-1169041-0
- Stéphane Louboutin, Calcul du nombre de classes des corps de nombres, Pacific J. Math. 171 (1995), no. 2, 455–467 (French, with French summary). MR 1372239
- Stéphane Louboutin, Class-number problems for cubic number fields, Nagoya Math. J. 138 (1995), 199–208. MR 1339949, DOI 10.1017/S0027763000005249
- Stéphane Louboutin, Class-group problems for cubic number fields, Japan. J. Math. (N.S.) 23 (1997), no. 2, 365–378. MR 1486518, DOI 10.4099/math1924.23.365
- Stéphane Louboutin, Computation of relative class numbers of imaginary abelian number fields, Experiment. Math. 7 (1998), no. 4, 293–303. MR 1678103
- S. Louboutin. Class number and class group problems for some non-normal totally real cubic number fields. In preparation.
- Franz Lemmermeyer and Attila Pethő, Simplest cubic fields, Manuscripta Math. 88 (1995), no. 1, 53–58. MR 1348789, DOI 10.1007/BF02567804
- C. Ricci. Ricerche arithmetiche sui polinomi. Rend. Circ. Mat. Palermo 57 (1933), 433-475.
- Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8
- Lawrence C. Washington, Class numbers of the simplest cubic fields, Math. Comp. 48 (1987), no. 177, 371–384. MR 866122, DOI 10.1090/S0025-5718-1987-0866122-8
Bibliographic Information
- Stéphane Louboutin
- Affiliation: Institut de Mathématiques de Luminy, UPR 906, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
- Email: loubouti@iml.univ-mrs.fr
- Received by editor(s): June 26, 2000
- Published electronically: June 8, 2001
- Communicated by: David E. Rohrlich
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 353-361
- MSC (1991): Primary 11R16, 11R29, 11R42
- DOI: https://doi.org/10.1090/S0002-9939-01-06168-8
- MathSciNet review: 1862112