Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T01:02:10.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Finding fundamental units in algebraic number fields

Published online by Cambridge University Press:  24 October 2008

Günter Lettl
Günter Lettl
Affiliation:
Institut für Mathematik, Karl-Franzens-Universität, Graz, Austria
Get access Rights & Permissions [Opens in a new window]

Extract

Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 383 - 388
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bantegnie, R.: Le ‘Problème des Octaédres’ en dimension 5. Acta Arithm. 14 (1968), 185–202.CrossRefGoogle Scholar
[2]Brunotte, H. and Halter-Koch, F.. Metrische Kennzeichnung von Erzeugenden für Einheitengruppen vom Rang 1 oder 2 in algebraischen Zahlkörpern. J. Number Theory 13 (1981), 320–333.CrossRefGoogle Scholar
[3]Cassels, J. W. S.. An Introduction to the Geometry of Numbers (Springer, 1959).CrossRefGoogle Scholar
[4]Cusick, T. W.. Finding fundamental units in totally real fields. Math. Proc. Cambridge Philos. Soc. 96 (1984), 191–194.CrossRefGoogle Scholar
[5]Halter-Koch, F.. Metrische Theorie der Einheiten algebraischer Zahlkörper. Mitt. Math. Ges. Hamburg 11 (1982), 131–141.Google Scholar
[6]Mordell, L. J.. Lattice octahedra. Canad. J. Math. 12 (1960), 297–302.CrossRefGoogle Scholar
[7]Wolff, K. H.. Über kritische Gitter im vierdimensionalen Raum (R4). Monatsh. Math. 58 (1954), 38–56.CrossRefGoogle Scholar