Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-20T18:50:06.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Kronecker classes of field extensions of small degree

Part of: Field extensions Permutation groups

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
Department of Mathematics University of Western AustraliaNedlands WA 6009, Australia
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.

The structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.

MSC classification

Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 12F10: Separable extensions, Galois theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 2 , April 1991 , pp. 297 - 315
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Cannon, J. J., ‘An introduction to the group theory language Cayley’, Computational group theory, edited by Atkinson, M., pp. 145–183, (Academic Press, New York, 1984).Google Scholar
[2]Guralnick, R. M., ‘Zeroes of permutation characters with applications to prime splitting and Brauer groups,’ (1988) preprint.Google Scholar
[3]Jehne, W., ‘Kronecker classes of algebraic number fields’, J. Number Theory 9 (1977), 279–320.CrossRefGoogle Scholar
[4]Jehne, W., ‘Kronecker classes of atomic extensions’, Proc. London Math. Soc. (3) 34 (1977), 32–64.CrossRefGoogle Scholar
[5]Klingen, N., ‘Zahlkörper mit gleicher Primzerlegung’, J. Reine Angew. Math. 229/300 (1978), 342–384.Google Scholar
[6]Klingen, N., ‘Atomare Kronecker-Klassen mit speziellen Galoisgruppen’, Abh. Math. Sem. Univ. Hamburg. 48 (1979), 42–53.CrossRefGoogle Scholar
[7]Klingen, N., ‘Rigidity of decomposition laws and number fields’, J. Austral. Math. Soc., Ser. A, (to appear).Google Scholar
[8]Liebeck, M. W., Praeger, C. E. and Saxl, J., ‘On the O'Nan-Scott Theorem for finite primitive permutation groups’, J. Austral. Math. Soc., Ser. A 44 (1988), 389–396.CrossRefGoogle Scholar
[9]Praeger, C. E., ‘Covering subgroups of groups and Kronecker classes of fields’, J. Algebra. 118 (1988), 455–463.CrossRefGoogle Scholar
[10]Praeger, C. E., ‘On octic extensions and a problem in group theory’, Group theory, Proceedings of the 1987 Singapore Conference, edited by Cheng, K. N. and Leong, Y. K., pp. 443–463, De Gruyter, (Berlin, New York, 1989).CrossRefGoogle Scholar
[11]Praeger, C. E., ‘On the inclusion problem for primitive permutation groups’, Proc. London Math. Soc. (3), 60 (1990), 68–88.CrossRefGoogle Scholar
[12]Saxl, J., ‘On a question of W. Jehne concerning covering subgroups of groups and Kronecker classes of fields’, Proc. London Math. Soc. (to appear).Google Scholar
[13]Sims, C. C., ‘Computational methods in the study of permutation groups’, Computational problems in abstract algebra, edited by Leech, J., pp. 169–183, (Pergamon Press, Oxford, London, 1970).Google Scholar