Cogalois and strongly Cogalois actions.

*(English)*Zbl 1170.20019Cogalois 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”.

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”.

Reviewer: Toma Albu (Bucureşti)

##### MSC:

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) |

##### Keywords:

abstract Cogalois theory; profinite groups; strong Cogalois groups; Kneser groups; radical groups; Kneser criterion; strong Cogalois actions
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\textit{Ş. A. Basarab}, J. Pure Appl. Algebra 212, No. 7, 1674--1694 (2008; Zbl 1170.20019)

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##### References:

[1] | Albu, T., (), 368 pp |

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[9] | Birkhoff, G.D.; Vandiver, H.S., On the integral divisors of \(a^n - b^n\), Ann. of math., 2, 5, 171-180, (1904) · JFM 35.0205.01 |

[10] | Macintyre, A., On definable subsets of \(p\)-adic fields, J. symbolic logic, 41, 605-610, (1976) · Zbl 0362.02046 |

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[12] | Serre, J.P., A course in arithmetic, (1973), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0256.12001 |

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