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An abstract cogalois theory for profinite groups. (English) Zbl 1155.12303
From the text: The aim of this paper is to develop an abstract group theoretic framework for the cogalois theory of field extensions, finite or not, which possess a cogalois correspondence. This theory is in a certain sense dual to the classical Galois theory dealing with field extensions possessing a Galois correspondence.
Let \(\Gamma\) be an arbitrary profinite group which acts continuously on the discrete subgroup \(A\) of the abelian group \(\mathbb Q/\mathbb Z\). The authors introduce the basic concepts of cogalois theory, Kneser and cogalois subgroups and the corresponding field extensions of \(Z'(\Gamma,A)\) of all continuous \(1\)-cocycles of \(\Gamma\) with coefficients in \(A\) in order to involve the group \(Z'(\Gamma,A)\) in defining the abstract concepts due to F. Barrera-Mora, M. Rzedowski-Calderón and G. D. Villa-Salvador [J. Pure Appl. Algebra 76, No. 1, 1–11 (1991; Zbl 0742.12001)]. Thereby they prove the abstract Kneser and quasi-purity criteria for Kneser and cogalois groups of cocycles.

12F10 Separable extensions, Galois theory
20E18 Limits, profinite groups
12G05 Galois cohomology
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06E15 Stone spaces (Boolean spaces) and related structures
Full Text: DOI
[1] Albu, T., Infinite field extensions with cogalois correspondence, Comm. algebra, 30, 2335-2353, (2002) · Zbl 1016.12003
[2] T. Albu, Cogalois Theory, A Series of Monographs and Textbooks, vol. 252, Marcel Dekker, New York, Basel, 2003, 368pp.
[3] Albu, T.; Nicolae, F., Kneser field extensions with cogalois correspondence, J. number theory, 52, 299-318, (1995) · Zbl 0838.12003
[4] Barrera-Mora, F.; Rzedowski-Calderón, M.; Villa-Salvador, G., On cogalois extensions, J. pure appl. algebra, 76, 1-11, (1991) · Zbl 0742.12001
[5] Johnstone, P.T., Stone spaces, (1982), Cambridge University Press Cambridge · Zbl 0499.54001
[6] Kneser, M., Lineare abhängigkeit von wurzeln, Acta arith., 26, 307-308, (1975) · Zbl 0314.12001
[7] Neukirch, J., Algebraische zahlentheorie, (1992), Springer Berlin, Heidelberg, New York · Zbl 0747.11001
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