A More General Framework for CoGalois Theory

Abstract

The coGalois theory studies the correspondence between subfields of a radical field extension L∕K and subgroups of the coGalois group coG(L∕K)-the torsion of the quotient group L ×∕K ×. Its abstract version concerns a continuous action of a profinite group Γ on a discrete quasi-cyclic group A, establishing a Galois connection between closed subgroups of Γ and subgroups of the group Z 1(Γ, A) of continuous 1-cocycles. More generally, we study in the present work triples \((\varGamma,\mathfrak{G},\eta )\), where Γ is a profinite group, \(\mathfrak{G}\) is a profinite operator Γ-group, and \(\eta:\varGamma \longrightarrow \mathfrak{G}\) is a continuous 1-cocycle such that η(Γ) topologically generates \(\mathfrak{G}\). To any such triple one assigns a natural coGalois connection between closed subgroups of Γ and closed Γ-invariant normal subgroups of \(\mathfrak{G}\). In the abelian context, examples concern the coGalois theory of separable radical extensions, an additive analogue of it based on Witt calculus and higher Artin-Schreier theory, and an extension of the abstract cyclotomic framework to Galois algebras. Kneser triples and coGalois triples are investigated, and general Kneser and coGalois criteria are provided. Problems on the classification of certain finite algebraic structures arising naturally from these criteria are stated and partial solutions are given.