I am studying galois theory and came across this idea. Suppose $L/K$ is a galois (?) extension of topological fields. Let $G$ be the group of continuous automorphisms of $L $ over $K$, equipped wirh krull topology.
I would like to show that there is a correspondence
$$ \{\text{closed subgroups of G} \} $$ $$ \leftrightarrow $$ $$ \{ \text{closed intermediate extensions } K \subset F \subset L \} $$
If it can be of any aid, one can try to show that if $K$ is a topological field, then Its algebraic closure $\bar K$ is so, and $K$ has the subspace topology.
It would be good to get a few examples of topological fields, so I ask:
- If $k$ is a tf, and $t$ a trascendent element, is $ k(t)$ a tf?
- What about if $ t $ is algebraic?
- A directed union of topological extensions is a topological extension. Thus from 1,2 would follow that any extension of a topological field is so, because it is the directed union of its finitely generated subextensions, which are towers of primitive extensions.
And now the final question. Suppose you have $L/K$ a galois extension of tfs with $ K \subset L$ closed, a convergent series $A=\sum a_n $ of elements $ a_n \in L $ such that for every $f \in G$, $f$ permutes the $a_n$. It is true that $A \in K$?
Thank you in advance! Best, Andrea