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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.

MSC:
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
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References:
[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
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