Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T08:16:03.714Z Has data issue: false hasContentIssue false
  • English
  • Français

On numbers which are differences of two conjugates over Q of an algebraic integer

Published online by Cambridge University Press:  17 April 2009

T. Zaimi
T. Zaimi
Affiliation:
King Saud University, Department of Mathematics, PO Box 2455, Riyadh 11451, Saudi Arabia
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 continue the investigation started by A. Dubickas of the numbers which are differences of two conjugates of an algebraic integer over the field Q of rational numbers. Mainly, we show that the cubic algebraic integers over Q with zero trace satisfy this property and we give a characterisation for those for which this property holds in their normal closure. We also prove that if a normal extension K/Q is of prime degree, then every integer of K with zero trace is a difference of two conjugates of an algebraic integer in K if and only if there exists an integer of K with trace 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 2 , October 2003 , pp. 233 - 242
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Dubickas, A., ‘On numbers which are differences of two conjugates of an algebraic integer’, Bull. Austral. Math. Soc. 65 (2002), 439–477.CrossRefGoogle Scholar
[2]Dubickas, A. and Smyth, C.J., ‘Variations on the theme of Hilbert's Theorem 90’, Glasgow Math. J. (to appear).Google Scholar
[3]Lang, S., Algebra (Addison-Wesley Publishing, Reading, MA, 1965).Google Scholar
[4]Schinzel, A., Selected topics on polynomials (University of Michigan, Ann Arbor, MI, 1982).CrossRefGoogle Scholar