# Galois theory for topological fields

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:

1. If $k$ is a tf, and $t$ a trascendent element, is $k(t)$ a tf?
2. What about if $t$ is algebraic?
3. 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

• Please use MathJax to format your posts: math.meta.stackexchange.com/questions/5020/… – jgon Jan 5 '18 at 19:50
• You’re thinking of extensions that are not of finite degree, I’m sure. My first question to you is whether they are algebraic extensions. My second question is how many specific examples you have found of such extensions $L\supset K$, and what phenomena you have observed so far. – Lubin Jan 5 '18 at 22:06
• Yes, they are algebraic extensions. – Andrea Marino Jan 6 '18 at 16:49
• About the second question, the second part is actually to find examples of such extensions! For example, it seems to me that C(s1,..., sn) in C(x1,..., xn) is closed and the galois group of permutations act continously (s_k being the elementary symmetric functions of x_j). – Andrea Marino Jan 6 '18 at 17:15