Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T02:19:27.854Z Has data issue: false hasContentIssue false
  • English
  • Français

An embedding theorem for fields

Published online by Cambridge University Press:  17 April 2009

J.W.S. Cassels
J.W.S. Cassels
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK.
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.

It is shown that every finitely generated field K of characteristic 0 may be embedded in infinitely many p-adic fields in such a way that the images of any given finite set C of non-zero elements of K are p-adic units. The result is suggested by Lech's proof of his generalization of Mahler's theorem on recurrent sequences. It also provides a simple proof of Selberg's theorem about torsion-free normal subgroups of matrix groups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 2 , April 1976 , pp. 193 - 198
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Borel, Armand, “Compact Clifford-Klein forms of symmetric spaces”, Topology 2 (1963), 111–122.CrossRefGoogle Scholar
[2]Borevich, Z.I. and Shafarevich, I.R., Number theory (translated by Newcomb Greenleaf for Scripta Technica. Pure and Applied Mathematics, 20. Academic Press, New York and London, 1966).Google Scholar
[3]Lech, Christer, “A note on recurring series”, Ark. Mat. 2 (1951), 417–421.CrossRefGoogle Scholar
[4]Mahler, Kurt, “Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen”, Proc. K. Nederl. Akad. Wetensch. Ameterdarn 38 (1935), 50–60.Google Scholar
[5]Mahler, K., “On the Taylor coefficients of rational functions”, Proc. Cambridge Philos. Soc. 52 (1956), 39–48.CrossRefGoogle Scholar
[6]Selberg, Atle, “On discontinuous groups in higher-dimensional symmetric spaces”, Contributions to function theory, 147–161 (International colloquim on Function Theory, Bombay, 1960. Tata Institute of Fundamental Research, Bombay, 1960).Google Scholar