I recently tried exploring a little bit of infinite Galois theory, although not in depth as i just wanted to build a few basic elements of the theory before going back to the classical finite theory.
Up until now i have defined the Krull Topology on a Galois extension E/K and given necessary and sufficient conditions on subsets of a group to be the neighbourhood system of the neutral element of that topological group. I have noticed that if E/K is a finite extension then we get the discrete topology on G.
I am not sure where to start with the following questions :
Let H be a subgroup of G with $K^H = K $. Show that for all Galois sub-extension L of E, finite over K, any K-automorphism of L is the restriction of an automorphism belonging to H.
By considering G a subset of $ \mathbb{N}^{ \mathbb{N}} $ reformulate this last result as " H is dense in G".
Show that the topology of $ G = Gal(E/K)$ is induced by the product topology of the discrete topologies on the factors of $ \mathbb{N}^{ \mathbb{N}} $.
Deduce from this that the topological group G is compact and totally disconnected.
If i am not mistaken, these questions enable us to study G without using the language of profinite groups. But how exactly, in simple terms, do I consider G as a subset of the sequence space $ \mathbb{N}^{ \mathbb{N}} $ ?