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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
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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 k0 so that k possesses non-trivial automorphisms leaving k0 pointwise fixed.

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