An Abstract Cogalois Theory for profinite groups

Abstract

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.

Field Theoretic Cogalois Theory via Abstract Cogalois Theory

“…In our paper [4] there was no indication about how the field theoretic results of Cogalois Theory can be deduced from their abstract group theoretic correspondents. This is done in the present paper.…”

“…First, observe that Γ E := Gal(Ω /E) = G ⊥ by Proposition 2.1 (4). Now, by [4, Corollary 2.14], G is a Γ -submodule of Z 1 (Γ , A) ⇐⇒ G ⊥ Γ ⇐⇒ E/F is a Galois extension.…”

“…The abstract correspondents for subgroups of Z 1 (Γ , A) of Kneser and Cogalois field extensions have been introduced in [4]. If one moves now via the maps (−) ⊥ from subgroups of Z 1 (Γ , A) to subgroups of the given profinite group Γ , then one can define the abstract versions for the latter of the concepts of radical, simple radical, Kneser, and Cogalois field extension.…”

“…where the left vertical arrow is the lattice isomorphism induced by ψ, the right vertical arrow is the canonical lattice anti-isomorphism E → Γ E := Gal(Ω /E) (with inverse ∆ → E ∆ ) given by the Fundamental Theorem of Infinite Galois Theory, the horizontal top arrows are the sup-semilattice morphism G → F(G) and the inf-semilattice morphism E → T (E/F), while the horizontal bottom arrows are the sup-semilattice anti-morphism G → G ⊥ and the inf-semilattice anti-morphism ∆ → ∆ ⊥ defined in [4]. The commutativity of the diagram above follows at once from [2, Theorem 1.8] or [3, Theorem 15.1.7].…”

“…This paper is a natural continuation of [4] were an abstract group theoretic framework of Cogalois Theory has been developed for arbitrary profinite groups. Cogalois Theory (see [3]) is a fairly recent domain of Field Theory investigating field extensions, finite or not, which possess a Cogalois correspondence.…”