Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T06:13:04.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields

Published online by Cambridge University Press:  17 April 2009

Elliot Benjamin
Elliot Benjamin
Affiliation:
Mathematics Department, Unity College, Unity ME 04988-9502, United States of America
Rights & Permissions [Opens in a new window]

Extract

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.

Let k be an imaginary quadratic number field and let k1 be the 2-Hilbert class field of k. If Ck,2, the 2-Sylow subgroup of the ideal class group of k, is elementary and |Ck,2|≥ 8, we show that Ck1,2 is not cyclic. If Ck,2 is isomorphic to Z/2Z × Z/4Z and Ck1,2 is elementary we show that k has finite 2-class field tower of length at most 2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 3 , December 1993 , pp. 379 - 383
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Blackburn, N., ‘On prime-power groups in which the derived group has two generators’, Proc. Camb. Phil. Soc. 53 (1957), 19–27.CrossRefGoogle Scholar
[2]Blackburn, N., ‘On prime-power groups with two generators’, Proc. Camb. Phil. Soc. 54 (1958), 327–337.CrossRefGoogle Scholar
[3]FurtwänglerPh,. Ph,., ‘Über die Klassenzahl Abelscher Zahlkörper’, Crelle 134 (1908), 91–94.CrossRefGoogle Scholar
[4]Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
[5]Hall, M. and Senior, J.K., The groups of order 2n(n≤6) (Macmillan, New York, 1964).Google Scholar
[6]Kisilevsky, H., ‘Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94’, J. Number Theory 8 (1976), 271–279.CrossRefGoogle Scholar
[7]Snyder, C. and Benjamin, E., ‘Number fields with 2-class group isomorphic to (2,2m)’, submitted for publication (1992).Google Scholar
[8]Taussky, O., ‘A remark on the class field tower’, J. London Math. Soc. 12 (1937), 82–85.Google Scholar