Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-09-03T17:23:00.902Z Has data issue: false hasContentIssue false
  • English
  • Français

The Construction of Fields with Infinite Cyclic Automorphism Group

Published online by Cambridge University Press:  20 November 2018

Willem Kuyk
Willem Kuyk*
Affiliation:
Mathematical Centre, Amsterdam, and McGill University, Montreal
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.

This paper deals with a problem raised in a paper by J. de Groot (1): Do there exist fields Ω whose full automorphism group is isomorphic to the additive group of integers Z?

The answer to this question is yes. In this paper we construct, given any subfield k of the complex numbers, extension fields Ω of k such that the automorphism group G(Ω/k) of Ω with respect to k is infinite cyclic. Fields having the infinite cyclic group as a full group of automorphisms are obtained by choosing the base field k in such a way that it does not contain any subfield k 0 so that k possesses non-trivial automorphisms leaving k 0 pointwise fixed.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 665 - 668
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. de Groot, J., Groups represented by homeomorphism groups I, Math. Ann., 138 (1959), 80–102.Google Scholar
2. Van der Waerden, B. L., Algebra, Vol. I (Berlin, 1955).Google Scholar