Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T23:58:42.510Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the characterization of CM-fields

Published online by Cambridge University Press:  09 April 2009

P. E. Blanksby  and
J. H. Loxton
P. E. Blanksby
Affiliation:
Department of Pure Mathematics The University of AdelaideAdelaideSouth Australia5001
J. H. Loxton
Affiliation:
School of Mathematics University of New South WalesKensingtonNew South Wales2033
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.

This note deals with some properties of algebraic number fields generated by numbers having all their conjugates on a circle. In particular, it is shown that an algebraic number field is a CM-field if and only if it is generated over the rationals by an element (not equal to ±1) whose conjugate all lie on the unit circle.

Subject classification (Amer. Math. Soc. (MOS) 1970): 12 A 15, 12 A 40, 14 K 22.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 1 , August 1978 , pp. 26 - 30
Copyright
Copyright © Australian Mathematical Society 1978

References

Ennola, V. and Smyth, C. J. (1974), “Conjugate algebraic numbers on a circle”, Annales Acad. Scientiarum Fennicae, Ser. A 582, 1–31.Google Scholar
Gold, R. (1974), “The non-triviality of certain Zl-extensions”, J. Number Theory 6, 369–373.CrossRefGoogle Scholar
Grossman, E. H. (1976), “On the solutions of diophantine equations in units”, Acta Arith. 30, 137–143.CrossRefGoogle Scholar
Györy, K. (1975), “Sur une classe des corps de nombres algébriques et ses applications”, Publ. Math. Debrecen 22, 151–175.CrossRefGoogle Scholar
Parry, C. J. (1975), “Units of algebraic number fields”, J. Number Theory 7, 385–388.CrossRefGoogle Scholar
Shimura, G. (1971), Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Math. Soc. Japan 11 (Princeton University Press).Google Scholar