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Upper and lower fields for profinite groups of given cardinality
Published online by Cambridge University Press: 09 April 2009
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Profinite groups (that is, compact totally disconnected topological groups) have been characterized as Galois groups in Krakowski (1971) and Leptin (1955). They can, in fact, be realized as groups of permutations on sets of transcendentals where every transcendental has a finite orbit under the group action and the fixed field is generated by the invariant rational functions on these orbits. If G = {G←x ¦ α∈A} we define the cardinality of G as ¦ G¦ = ¦A¦. In this paper we construct, for every cardinal number c and profinite group G with ¦G¦ ≦ c two universal fields kc and Kc so that G ≃ Gal (KcKG) and G ≃ (KGKc) for fields KG and kG which depend on G. See Cassels and Frohlich (1967) for a description of the basic properties of profinite groups.
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- Copyright © Australian Mathematical Society 1976
References
Cassels, J. W. S., and Frohlich, A. (1967), Algebraic Number Theory , Chapter V; (Thompson Book Co. Washington D. C. 1967).Google Scholar
Jacobson, N. (1964), Lectures in Abstract Algebra. Vol. III, 1964, (D. Van Nostrand Co., Princeton New Jersey, 1964).Google Scholar
Krakowski, D., Profinite Groups and the Galois Groups of Fields, (Thesis), (submitted to Univ. of Cal. at Los Angeles, 06 1971).Google Scholar
Leptin, H. (1955), ‘Compact, totally disconnnected groups’, Arch. Math. 6 371–373.CrossRefGoogle Scholar
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