  The paper is an extended version of a talk given at the International
Conference: Experimental and Theoretical Methods in Algebra, Geometry and
Topology, which took place June 21--24, 2013 in Eforie Nord (Romania). Its
purpose is to present a more general framework for a fairly new theory in Field
Theory, called coGalois Theory, which is somewhat dual to the very classical
Galois Theory and is more general than the Kummer Theory. The main object of
investigation in this more general framework is the coGalois connexion
naturally associated to any triple $\,(\Gamma, \frak{G}, \eta)\,$, where
$\,\Gamma\,$ is a profinite group, $\,\frak{G}\,$ is a profinite operator
$\,\Gamma$-group, and $\,\eta: \Gamma \longrightarrow \frak{G}\,$ is a
continuous $\,1$-cocycle with the property that the profinite group
$\,\frak{G}\,$ is topologically generated by the image $\,\eta(\Gamma)$.
