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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:
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