×

zbMATH — the first resource for mathematics

The non-normal quartic CM-fields and the octic dihedral CM-fields with relative class number two. (English) Zbl 0976.11051
S. Louboutin and R. Okazaki have determined exactly all 38 non-isomorphic, non-normal quartic CM-fields with relative class number one and exactly all 19 non-isomorphic octic dihedral CM-fields with relative class number one [Acta Arith. 67, 47-62 (1994; Zbl 0809.11069)].
In this paper, the author intends to obtain analogous results for relative class number two and determines exactly all 254 non-isomorphic, non-normal quartic CM-fields with relative class number two and exactly all 95 non-isomorphic octic dihedral CM-fields with relative class number two. Moreover, by using this fact he determines precisely all 16 non-isomorphic octic dihedral CM-fields the ideal class groups of which the are nontrivial cyclic of 2-power order.

MSC:
11R29 Class numbers, class groups, discriminants
11R21 Other number fields
Software:
KANT/KASH
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Daberkow, M.; Fieker, C.; Klüners, J.; Pohst, M.; Roegner, R.; Wildanger, K.: Kant v4. J. symbolic comput. 24, 267-283 (1997) · Zbl 0886.11070
[2] Kubota, T.: Über den bizyklischen biquadratischen zalhkörper. Nagoya math. J. 10, 65-85 (1956) · Zbl 0074.03001
[3] Lemmermeyer, F.: Ideal class groups of cyclotomic number fields, I. Acta arith. 27, 347-359 (1995) · Zbl 0837.11059
[4] Louboutin, S.: On the class number one problem for non-normal quartic CM-fields. Tôhoku math. J. 46, 1-12 (1994) · Zbl 0796.11050
[5] Louboutin, S.: The class number one problem for the non-abelian normal CM-fields of degree 16. Acta arith. 82, 173-196 (1997) · Zbl 0881.11079
[6] Louboutin, S.: CM-fields with cyclic ideal class groups of 2-power orders. J. number theory 67, 1-10 (1997) · Zbl 0881.11078
[7] Louboutin, S.: Computation of relative class numbers of CM-fields. Math. comp. 66, 1185-1194 (1997) · Zbl 0879.11064
[8] S. Louboutin, Hasse unit indices of dihedral octic CM-fields, Math. Nachr, in press. · Zbl 0972.11105
[9] Louboutin, S.: Continued fractions and real quadratic fields. J. number theory 30, 167-176 (1988) · Zbl 0652.12002
[10] Louboutin, S.: Calcul du nombre de classes des corps de nombres. Pacific J. Math. 171, 455-467 (1995) · Zbl 0854.11060
[11] Louboutin, S.: Groupes des classes d’idéaux triviaux. Acta arith. 54, 61-74 (1989) · Zbl 0634.12008
[12] Louboutin, S.; Okazaki, R.: Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta arith. 67, 47-62 (1994) · Zbl 0809.11069
[13] C. Batut, D. Bernardi, H. Cohen, and, M. Olivier, PARI system, version 1.38, 1983.
[14] Serre, J. P.: Représentation linéaire des groups finis. (1982)
[15] Washington, L. C.: Introduction to cyclotomic fields. Grad texts in math. 83 (1997) · Zbl 0966.11047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.