Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-18T23:40:30.668Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigidity of Decomposition laws and number fields

Part of: Algebraic number theory: global fields Permutation groups

Published online by Cambridge University Press:  09 April 2009

Norbert Klingen
Norbert Klingen
Affiliation:
Mathematiches Institut Universität zu KölnWeyertal 86-90 D 5000 Köln 41, Germany
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 speak of rigidity, if partial information about the prime decomposition in an extension of number fields K¦k determines the decomposition law completely (and hence the zeta function ζK), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

MSC classification

Secondary: 11R32: Galois theory 11R27: Units and factorization 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 171 - 186
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Artin, E. and Tate, J., Class field theory, (Benjamin, Reading, Mass., 1967).Google Scholar
[2]Bauer, M., ‘Über Kreisteilungsgleichungen’, Arch. Math. Phys. 6 (1904), 220.Google Scholar
[3]Bauer, M., ‘Zur Theorie der algebraischen Zahlkörper’, Math. Ann. 77 (1916), 353–356.CrossRefGoogle Scholar
[4]Brandl, R., Letter to Jehne, W., 04 22, 1983.Google Scholar
[5]Butler, G. and McKay, J., ‘The transitive groups of degree up to eleven’, Comm. Algebra 11 (1983), 863–911.CrossRefGoogle Scholar
[6]Cassels, J. and Fröhlich, A. (eds.), Algebraic number theory, (Academic Press, London, New York, 1967).Google Scholar
[7]Gaßmann, F., ‘Bemerkungen zur vorstheenden Arbeit von Hurwitz’, Math. Z. 25 (1926), 665–675.Google Scholar
[8]Gerst, I., ‘On the theory of nth power residues and a conjecture of Kronecker’, Acta Arith. 17 (1970), 121–139.Google Scholar
[9]Huppert, B., Endliche Gruppen I, (Springer, Berlin, Heidelberg, New York, 1967).Google Scholar
[10]Jehne, W., ‘Kronecker classes of algebraic number fields’, J. Number Theory 9 (1977), 279–320.Google Scholar
[11]Klingen, N., ‘Zahlkörper mit gleicher Primzerlegung’, J. Reine Angew. Math. 299/300 (1978), 342–384.Google Scholar
[12]Klingen, N., ‘Atomare Kronecker-Klassen mit speziellen Galoisgruppen’, Abh. Math. Sem. Univ. Hamburg 48 (1979), 42–53.CrossRefGoogle Scholar
[13]Klingen, N., ‘Über schwache quadratische Zerlegungsgesetze’, Comment. Math. Helv. 55 (1980), 645–651.CrossRefGoogle Scholar
[14]Komatsu, K., ‘On the adele rings and zeta-functions of algebraic number fields’, Kodai Math. J. 1 (1978), 394–400.Google Scholar
[15]McKay, J., ‘Some remarks on computing Galois groups’, SIAM J. Comput. 8 (1979), 344–347.CrossRefGoogle Scholar
[16]Neukirch, J., Class field theory, (Grundlehren Math. Wiss. 280, Springer, Berlin, Heidelberg, New York, Tokyo, 1986).Google Scholar
[17]O'Meara, O. T., Introduction to quadratic forms, (Grundlehren Math. Wiss. vol. 117, Springer-Verlag, Göttingen, Heidelberg, 1963).CrossRefGoogle Scholar
[18]Perlis, R., ‘On the equation ζK(s) = ζK1(S)’, J. Number Theory 9 (1977), 342–360.CrossRefGoogle Scholar
[19]Praeger, C. E., ‘Covering subgroups of groups and Kronecker classes of fields’, J. Algebra, 118 (1988), 455–463.Google Scholar
[20]Praeger, C. E., ‘Kronecker classes of field extensions of small degree’, J. Austral. Math. Soc. Ser. A 50 (1991), 297–315.CrossRefGoogle Scholar
[21]Saxl, J., ‘On a question of W. Jehne concerning covering subgroups of groups and Kronecker classes of fields’, J. London Math. Soc., to appear.Google Scholar
[22]Schinzel, A., ‘On a theorem of Bauer and some of its applications’, Acta Arith. 11 (1966), 333–344.CrossRefGoogle Scholar
[23]Serre, J.-P., ‘L'invariant de Witt de la forme Tr(x2)’, Comment. Math. Helv. 59 (1984), 651–676.Google Scholar
[24]Tninks, W., Arithmetisch ähnliche Körper, (Diplomarbeit, Karlsruhe, 1969).Google Scholar