TITLE: Galois actions in towers and class field theory
QUESTION [6 upvotes]: I'm trying to understand how this works in terms of Galois theory and local class field theory. Assume we have an extension of local fields $E/L/K$ s.t. $E/L$ and $L/K$ are abelian. I'm interested in recognizing when $E/K$ is Galois. Clearly, $E/K$ is Galois if and only if $E$ is always fixed by an extension of an $L/K$ automorphism to $E$, but this is tricky to compute.
I overheard a brief conversation that this can be done through Galois groups and some group actions that occur in the tower, but I haven't found anything explicit through google. We should be able to see from how $Gal(L/K)$ acts on something whether or not the extension is Galois. I'm having trouble seeing what the action should be. I hope someone who knows what I'm talking about could write it down explicitly. Since the Galois groups should correspond through local class field theory to very concrete objects which are quotients of $E^\times$, $L^\times$ and $K^\times$. I was wondering how this action on the Galois side is expressed on the local field side?
I'm interested in this since it clearly would provide a tool for constructing some solvable extensions of e.g. $\mathbb{Q}_p$. I apologize for being fuzzy, but I don't know how to be more explicit.

REPLY [2 votes]: If $L/K$ is abelian and $K/k$ cyclic, then $L/k$ is going to be normal if the Galois group at the gottom acts on the class group attached to $L/K$ via class field theory, and it will be abelian if action fixes the classes. This is due to Hasse and follows easily from the standard results in class field theory. This is Prop. 1.2.8 in my survey on class field towers (I apologize for the outdated content. All of this needs to be rewritten).
There are similar constructions in Kummer theory going back to Kummer; this can be used in several proofs of the Kronecker-Weber theorem and should be e.g. in Washington's book; see also Prop. 2.2.1 in the survey
If the base extension is abelian and not cyclic, stuff happens. There are many articles investigating this problem, but nothing as simple as in the cyclic case.