Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T23:56:03.507Z Has data issue: false hasContentIssue false
  • English
  • Français

NONABELIAN NORMAL CM-FIELDS OF DEGREE 2pq

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  01 August 2009

S.-H. KWON ,
S. LOUBOUTIN  and
S.-M. PARK
S.-H. KWON
Affiliation:
Department of Mathematics Education, Korea University, 136-701, Seoul, Korea (email: [email protected])
S. LOUBOUTIN*
Affiliation:
Institut de Mathématiques de Luminy, UMR 6206, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected])
S.-M. PARK
Affiliation:
Department of Mathematics, Korea University, 136-701, Seoul, Korea (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that the relative class number of a nonabelian normal CM-field of degree 2pq (where p and q are two distinct odd primes) is always greater than four. Not only does this result solve the class number one problem for the nonabelian normal CM-fields of degree 42, but it has also been used elsewhere to solve the class number one problem for the nonabelian normal CM-fields of degree 84.

Keywords

CM-fieldrelative class numbernumber field

MSC classification

Secondary: 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 1 , August 2009 , pp. 129 - 144
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

The research of the first author was supported by a Korea University Grant.

References

[1]Batut, C., Belabas, K., Bernardi, D., Cohen, H. and Olivier, M., PARI-GP version 2.1.5,http://pari.math.u-bordeaux.fr.Google Scholar
[2]Bessassi, S., ‘Bounds for the degrees of CM-fields of class number one’, Acta Arith. 106 (2003), 213–245.CrossRefGoogle Scholar
[3]Chang, K.-Y. and Kwon, S.-H., ‘Class number problem for imaginary cyclic number fields’, J. Number Theory 73 (1998), 318–338.CrossRefGoogle Scholar
[4]Chang, K.-Y. and Kwon, S.-H., ‘Class numbers of imaginary abelian fields’, Proc. Amer. Math. Soc. 128 (2000), 2517–2528.Google Scholar
[5]Chang, K.-Y. and Kwon, S.-H., ‘The nonabelian CM-fields of degree 36 with class number one’, Acta Arith. 101 (2002), 53–61.CrossRefGoogle Scholar
[6]Cox, D. A., ‘Primes of the form x 2+ny 2’, in: Fermat, Class Field Theory, and Complex Multiplication (John Wiley & Sons, New York, 1989).Google Scholar
[7]Daberkow, M., Fieker, C., Klüners, J., Pohst, M., Roegner, K., Schörnig, M. and Wildanger, K., ‘KANT V 4’, J. Symbolic Comput. 24 (1997), 267–283.Google Scholar
[8]Hida, H., Elementary Theory of L-Functions and Eisenstein Series, London Mathematical Society, Student Texts, 26 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
[9]Kwon, S.-H. and Martinet, J., ‘Sur les corps résolubles de degré premier’, J. Reine. Angew. Math. 375 (1987), 12–23.Google Scholar
[10]Lee, G.-N. and Kwon, S.-H., ‘CM-fields with relative class number one’, Math. Comp. 75 (2006), 997–1013.Google Scholar
[11]Lefeuvre, Y., ‘Corps diédraux à multiplication complexe principaux’, Ann. Inst. Fourier 50 (2000), 67–103.Google Scholar
[12]Lefeuvre, Y. and Louboutin, S., ‘The class number one problem for the dihedral CM-fields’, in: Algebraic Number Theory and Diophantine Analysis (Graz, 1998) (de Gruyter, Berlin, 2000), pp. 249–275.Google Scholar
[13]Louboutin, S., ‘Minoration au point 1 des fonctions L et détermination des corps sextiques abéliens totalement imaginaires principaux’, Acta Arith. 62 (1992), 109–124.CrossRefGoogle Scholar
[14]Louboutin, S., ‘Upper Bounds on ∣L(1,χ)∣ and applications’, Canad. J. Math. 50 (1998), 794–815.CrossRefGoogle Scholar
[15]Louboutin, S., ‘The class number one problem for the dihedral and dicyclic CM-fields’, Colloq. Math. 80 (1999), 259–265.Google Scholar
[16]Louboutin, S., ‘Computation of relative class numbers of CM-fields by using HeckeL-functions’, Math. Comp. 69 (2000), 371–393.CrossRefGoogle Scholar
[17]Louboutin, S., ‘Computation of L(0,χ) and of relative class numbers of CM-fields’, Nagoya Math. J. 161 (2001), 171–191.CrossRefGoogle Scholar
[18]Louboutin, S., ‘Explicit lower bounds for residues at s=1 of Dedekind zeta functions and relative class numbers of CM-fields’, Trans. Amer. Math. Soc. 355 (2003), 3079–3098.CrossRefGoogle Scholar
[19]Louboutin, S. and Okazaki, R., ‘Determination of all nonnormal quartic CM-fields and of all nonabelian normal octic CM-fields with class number one’, Acta Arith. 67 (1994), 47–62.CrossRefGoogle Scholar
[20]Louboutin, S. and Okazaki, R., ‘The class number one problem for some nonabelian normal CM-fields of 2-power degrees’, Proc. London Math. Soc. 76 (1998), 523–548.Google Scholar
[21]Louboutin, S., Okazaki, R. and Olivier, M., ‘The class number one problem for some nonabelian normal CM-fields’, Trans. Amer. Math. Soc. 349 (1997), 3657–3678.CrossRefGoogle Scholar
[22]Louboutin, S., Park, Y.-H. and Lefeuvre, Y., ‘Construction of the real dihedral number fields of degree 2p. Applications’, Acta Arith. 89 (1999), 201–215.CrossRefGoogle Scholar
[23]Odlyzko, A., ‘Some analytic estimates of class number and discriminants’, Invent. Math. 29 (1975), 275–286.CrossRefGoogle Scholar
[24]Okazaki, R., ‘Inclusion of CM-fields and divisibility of relative class numbers’, Acta Arith. 92 (2000), 319–338.CrossRefGoogle Scholar
[25]Park, Y.-H. and Kwon, S.-H., ‘Determination of all imaginary abelian sextic number fields with class number ≤11’, Acta Arith. 82 (1997), 27–43.CrossRefGoogle Scholar
[26]Park, Y.-H. and Kwon, S.-H., ‘Determination of all nonquadratic imaginary cyclic number fields of 2-power degree with class number ≤20’, Acta Arith. 83 (1998), 211–223.CrossRefGoogle Scholar
[27]Park, S.-M. and Kwon, S.-H., ‘Class number one problem for normal CM-fields’, J. Number Theory 125 (2007), 59–84.CrossRefGoogle Scholar
[28]Stark, H., ‘A complete determination of all complex quadratic fields of class number one’, Michigan Mathematics J. 14 (1967), 1–27.CrossRefGoogle Scholar
[29]Washington, L. C., Introduction to Cyclotomic Fields, 2nd edn, Graduate Texts in Mathematics, 83 (Springer, Berlin, 1997).CrossRefGoogle Scholar
[30]Watkins, M., ‘Real zeros of real odd Dirichlet L-functions’, Math. Comp. 73 (2004), 415–423.CrossRefGoogle Scholar
[31]Watkins, M., ‘Class numbers of imaginary quadratic fields’, Math. Comp. 73 (2004), 907–938.Google Scholar
[32]Yamamura, K., ‘The determination of the imaginary abelian number fields with class number one’, Math. Comp. 62 (1994), 899–921.Google Scholar