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The Automorphisms of an Algebraically Closed Field

Published online by Cambridge University Press:  20 November 2018

A. Charnow
A. Charnow*
Affiliation:
California State College, Hayward, California
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It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k.

The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 95 - 97
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Adamson, I., Introduction to field theory, Oliver and Boyd, London, 1964.Google Scholar
2. Jacobson, N., Lectures in abstract algebra, vol. 3, Van Nostrand, Princeton, N.J., 1964.Google Scholar