The class number one problem for some non-abelian normal CM-fields of degree
F. Lemmermeyer; S. Louboutin; R. Okazaki
Journal de théorie des nombres de Bordeaux (1999)
- Volume: 11, Issue: 2, page 387-406
- ISSN: 1246-7405
Access Full Article
topAbstract
topHow to cite
topLemmermeyer, F., Louboutin, S., and Okazaki, R.. "The class number one problem for some non-abelian normal CM-fields of degree $24$." Journal de théorie des nombres de Bordeaux 11.2 (1999): 387-406. <http://eudml.org/doc/248333>.
@article{Lemmermeyer1999,
abstract = {We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to $\mathcal \{A\}_4$, the alternating group of degree $4$ and order $12$. There are two such fields with Galois group $\mathcal \{A\}_4 \times \mathcal \{C\}_2$ (see Theorem 14) and at most one with Galois group SL$_2(\mathbb \{F\}_3)$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$.},
author = {Lemmermeyer, F., Louboutin, S., Okazaki, R.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {nonabelian normal CM-fields; class number one; Galois group},
language = {eng},
number = {2},
pages = {387-406},
publisher = {Université Bordeaux I},
title = {The class number one problem for some non-abelian normal CM-fields of degree $24$},
url = {http://eudml.org/doc/248333},
volume = {11},
year = {1999},
}
TY - JOUR
AU - Lemmermeyer, F.
AU - Louboutin, S.
AU - Okazaki, R.
TI - The class number one problem for some non-abelian normal CM-fields of degree $24$
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 2
SP - 387
EP - 406
AB - We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to $\mathcal {A}_4$, the alternating group of degree $4$ and order $12$. There are two such fields with Galois group $\mathcal {A}_4 \times \mathcal {C}_2$ (see Theorem 14) and at most one with Galois group SL$_2(\mathbb {F}_3)$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$.
LA - eng
KW - nonabelian normal CM-fields; class number one; Galois group
UR - http://eudml.org/doc/248333
ER -
References
top- [1] J.V. Armitage, A. Frohlich, Classnumbers and unit signatures. Mathematika14 (1967), 94-98. Zbl0149.29501MR214566
- [2] C. Bachoc, S.-H. Kwon, Sur les extensions de groupe de Galois Ã4. Acta Arith.62 (1992), 1-10. Zbl0784.11051MR1179006
- [3] H. CohenA course in computational algebraic number theory. Grad. Texts Math. 138, Springer-Verlag, Berlin, Heidelberg, New York (1993). Zbl0786.11071MR1228206
- [4] H. Cohen, F. Diaz y Diaz, M. Olivier, Tables of octic fields with a quartic subfield. Math. Comp., à paraître. Zbl1036.11066
- [5] A. Derhem, Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques. thèse, Université Laval, Québec (1988).
- [6] Ph. Furtwängler, Über das Verhalten der Ideale des Grundkörpers im Klassenkörper. Monatsh. Math. Phys.27 (1916), 1-15. Zbl46.0246.01MR1548760JFM46.0246.01
- [7] H. Heilbronn, Zeta-functions and L-functions. Algebraic Number Theory, Chapter VIII, J.W.S. Cassels and A. Fröhlich, Academic Press, (1967). MR218327
- [8] G. James, M. Liebeck, Representations and Characters of groups. Cambridge University Press, (1993). Zbl0792.20006MR1237401
- [9] KANT V4, by M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, J. Symbolic Computation24 (1997),267-283. Zbl0886.11070MR1484479
- [10] H. Koch, Über den 2-Klassenkörperturm eines quadratischen Zahlkörpers. I, J. Reine Angew. Math.214/215 (1963), 201-206. Zbl0123.03904MR164945
- [11] P. Kolvenbach, Zur arithmetischen Theorie der SL(2, 3)-Erweiterungen. Diss. Köln, 1982.
- [12] S.-H. Kwon, Corps de nombres de degré 4 de type alterné. C. R. Acad. Sci. Paris299 (1984), 41-43. Zbl0564.12007MR756515
- [13] S. Lang, Cyclotomic Fields II. Springer-Verlag1980 Zbl0435.12001MR566952
- [14] F. Lemmermeyer, Ideal class groups of cyclotomic number fields I. Acta Arith.72 (1995), 347-359. Zbl0837.11059MR1348202
- [15] F. Lemmermeyer, Unramified quaternion extensions of quadratic number fields. J. Théor. N. Bordeaux9 (1997), 51-68. Zbl0890.11031MR1469661
- [16] Y. Lefeuvre, Corps diédraux à multiplication complexe principaux. Ann. Inst. Fourier, à paraitre. Zbl0952.11024
- [17] S. Louboutin, R. Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta Arith.67 (1994), 47-62. Zbl0809.11069MR1292520
- [18] S. Louboutin, R. Okazaki, The class number one problem for some non-abelian normal CM-fields of 2-power degrees. Proc. London Math. Soc.76, No.3 (1998), 523-548. Zbl0891.11054MR1616805
- [19] S. Louboutin, R. Okazaki, M. Olivier, The class number one problem for some non-abelian normal CM-fields. Trans. Amer. Math. Soc.349 (1997), 3657-3678. Zbl0893.11045MR1390044
- [20] S. Louboutin, Lower bounds for relative class numbers of CM-fields. Proc. Amer. Math. Soc.120 (1994), 425-434. Zbl0795.11058MR1169041
- [21] S. Louboutin, Majoration du résidu au point 1 des fonctions zêta de certains corps de nombres. J. Math. Soc. Japan50 (1998), 57-69. Zbl1040.11081MR1484611
- [22] S. Louboutin, The class number one problem for the non-abelian normal CM-fields of degree 16. Acta Arith.82 (1997), 173-196. Zbl0881.11079MR1477509
- [23] S. Louboutin, Upper bounds on |L(1, χ)| and applications. Canad. J. Math., 50 (4), (1998), 794-815. Zbl0912.11046
- [24] R. Okazaki, Inclusion of CM-fields and divisibility of class numbers. Acta Arith., à paraître. Zbl0952.11023
- [25] User's Guide to PARI-GP, by C. Batut,K. Belabas,D. Bernardi,H. CohenandM. Olivierlast updated Nov. 14, 1997; Laboratoire A2X, Univ. Bordeaux I.
- [26] Y.-H. Park, S.-H. Kwon, Determination of all imaginary abelian sextic number fields with class number < 11. Acta Arith.82 (1997), 27-43. Zbl0889.11036MR1475764
- [27] A. Rio, Dyadic exercises for octahedral extensions. preprint 1997; URL: http://www-ma2.upc.as/~rio/papers.html Zbl0941.12001
- [28] H.M. Stark, Some effective cases of the Brauer-Siegel theorem. Invent. Math.23 (1974), 135-152. Zbl0278.12005MR342472
- [29] O. Taussky, A remark on the class field tower. J. London Math. Soc.12 (1937), 82-85. Zbl0016.20002JFM63.0144.03
- [30] A.D. Thomas, G.V. Wood, Group Tables, Shiva Publishing Ltd, Kent, UK 1980. Zbl0441.20001MR572793
- [31] L.C. Washington, Introduction to Cyclotomic Fields. Grad. Texts Math. 83, Springer-Verlag1982; 2nd edition 1997. Zbl0484.12001MR718674
- [32] A. Weil, Exercices dyadiques. Invent. Math.27 (1974), 1-22. Zbl0307.12017MR379445
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.