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Cogalois and strongly Cogalois actions. (English) Zbl 1170.20019
Cogalois Theory, a fairly new area in Field Theory, investigates field extensions, finite or not, that possess a so called ‘Cogalois correspondence’ [see the reviewer’s monograph Cogalois Theory, (Pure Appl. Math. 252), New York: Marcel Dekker (2003; Zbl 1039.12001)]. The subject is somewhat dual to the very classical ‘Galois Theory’ dealing with field extensions possessing a ‘Galois correspondence’.
In 2005 the (field theoretic) Cogalois Theory has been generalized to arbitrary profinite groups by T. Albu and Ş. A. Basarab, [J. Pure Appl. Algebra 200, No. 3, 227–250 (2005; Zbl 1155.12303)], leading to a so called ‘Abstract Cogalois Theory’ for arbitrary profinite groups.
The aim of the paper under review is to continue the development of the Abstract Cogalois Theory initiated in the paper mentioned above and further studied in the author’s paper [Serdica Math. J. 30, No. 2-3, 325–348 (2004; Zbl 1066.20032)]. Thus, the author investigates ‘Cogalois’ and ‘strongly Cogalois’ subgroups of a profinite group \(\Gamma\) acting continuously on a subgroup \(A\) of \(\mathbb Q/\mathbb Z\), and introduces and completely classifies two types of actions called ‘Cogalois actions’ and ‘strongly Cogalois actions’. The paper is very technical, so it is difficult to explain all its new concepts and results.
Note that original term of “Cogalois” is spelled in the paper under review as “coGalois”, while in other papers in the literature it is spelled as “co-Galois”.

20E18 Limits, profinite groups
12F05 Algebraic field extensions
12F10 Separable extensions, Galois theory
12G05 Galois cohomology
06A15 Galois correspondences, closure operators (in relation to ordered sets)
Full Text: DOI
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