The aim of this paper is to develop an abstract group theoretic framework for the Cogalois Theory of field extensions.The efforts to generalize the famous Gauss'Quadratic Reciprocity Law led to the Theory of Abelian extensions of global and local fields, known as Class Field Theory. This Theory can also be developed in an abstract group theoretic framework, namely for arbitrary profinite groups. Since the profinite groups are precisely those topological groups which arise as Galois groups of Galois extensions, an Abstract Galois Theory for arbitrary profinite groups was developed within the Abstract Class Field Theory (see e.g. [7]).The aim of this paper is to present a dual Theory, we called Abstract Cogalois Theory, to the Abstract Galois Theory. Roughly speaking, Cogalois Theory (see [2]) investigates field extensions, finite or not, which possess a Cogalois correspondence. This Theory is somewhat dual to the very classical Galois Theory dealing with field extensions possessing a Galois correspondence.
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