Galois actions in towers and class field theory 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.
 A: When you have a finite abelian extension $E|L$ of a finite galoisian extension $L|K$ of local fields, the extension $E|K$ is going to be galoisian if and only if the subgroup $N_{E|L}(E^\times)\subset L^\times$ is $Gal(L|K)$-stable.
This is somewhat similar to the following purely algebraic fact :  Let $K$ be any field, $L$ a finite galoisian extension of $K$ of group $G=\mathrm{Gal}(L|K)$, and suppose that $L^\times$ has an element of order $n$ for some $n>1$.  Then abelian extensions $E$ of $L$ of exponent dividing $n$ correspond bijectively to subgroups $H$ of $L^\times/L^{\times n}$ under the map $E\mapsto\mathrm{Ker}(L^\times/L^{\times n}\to E^\times/E^{\times n})$, the reciprocal being $H\mapsto L(\root n\of H)$ (Kummer theory).  In such a situation, the abelian extension $E=L(\root n\of H)$ of $L$ is galoisian over $K$ if and only if the subgroup $H\subset L^\times/L^{\times n}$ is $G$-stable.
One has a similar (purely algebraic) statement in Artin-Schreier theory of abelian extension of exponent $p$ in characteristic $p$, and indeed in Witt's theory of abelian extensions of exponent dividing $p^n$.
A: Not an answer, but perhaps something useful from an apprentice in CFT:
Considerations very closely related to your question led Andre Weil to discover the "Weil group"; see his paper on Class Field theory (1955). 
Let $G$ be the Galois group of $L/K$ and $H$ the Galois group of $E/L$. 
The multiplicative group $L^*$is naturally a $G$-module and so one can consider the cohomology group $H^2(G, {L^*})$. Now $E^*$is not a $G$-module in any natural way. But it does give rise to a $G$-module which should be useful in answering your question.
Namely, the subgroup $NE^*$ of norms from $E$ to $L$ is a $G$-module. So one can consider the quotient $G$-module $M = {L^*}/{NE^*}$ and the cohomology group $H^2(G, M)$. 
If $E/K$ is Galois, then its Galois group $\Gamma$ would be sit in an exact sequence $$0 \to H \to \Gamma \to G \to 0.$$ The conjugation action of $\Gamma$ on its normal subgroup $H$ (identified with its image in $\Gamma$) factorises via the quotient ${\Gamma}/{H} = G$. So this gives $H$ a $G$-module structure. 
The group $\Gamma$ gives rise to an element $\gamma$ of $H^2(G, H)$ (via a standard construction). Local class field theory provides us with 


*

*a fundamental class $u_{L/K}$ in $H^2(G, {L^*})$, and

*an isomorphism $H \cong M$.


The latter yields a map $L^* \to M \cong H$ of $G$-modules which provides a map $$t: H^2(G, L^*) \to H^2(G, H).$$
If $E/K$ were Galois, then the Galois group $\Gamma$ would have a constraint, namely, $$\gamma = t(u_{L/K}).$$
This discussion presupposes that $E.K$ is Galois and gives a constraint; it does not answer your question on how to check if $E/K$ is Galois!!
The fundamental class is what gives rise to the local Weil group.
By the way, every finite Galois extension of $Q_p$ (or a $p$-adic local field) is solvable.
All of this can be found in Cassels-Frohlich or Serre's Local Fields or Milne's notes on Class Field Theory.
Let us wait for a master in LCFT for an answer!!
A: 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.
