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Assume we are given an extension of number fields or $\mathfrak{p}$-adic number fields $L/E/K$ where each extension is abelian and $L/K$ is only assumed Galois. Now take any element $\sigma\in \textrm{Gal}(E/K)$. We can extend $\sigma$ to an element of $\tilde{\sigma}\in \textrm{Gal}(L/K)$. If $\tilde{\eta}$ is another such extension, then $\tilde{\sigma}$ and $\tilde{\eta}$ are conjugate by an element of $\textrm{Gal}(L/E)$. It follows that $\textrm{Gal}(E/K)$ defines a well-defined action on $\textrm{Gal}(L/E)$ by conjugating with a lifting, since the result of the conjugation is independent of the particular lift.

My question is essentially the following: Will this action give us any useful information about the extension $L/K$? Does it fit into the functoriality of class field theory in some way? For example, taking $E/K$ to be quadratic and $L/E$ to be an abelian extension of degree $4$, then we have that the action is trivial if only if $L/K$ is also abelian, so it should let us distinguish between abelian extensions and non-abelian extensions. How far can this be pushed in general and how much information can we generally get out of it? Are there any other interesting actions induced by different groups appearing in class field theory?

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Yes, this action does match up with the natural action of $Gal(E/K)$ on the ray class groups of $E$. This idea can be pushed an awfully long way and in a sense mathematicians are still working on it today: the question of how $Gal(E/K)$ acts on the Galois groups of abelian extensions of $E$ is an important part of the field known as Iwasawa theory. (Iwasawa theorists particularly consider the case where $E/K$ is an infinite Galois extension, in which case the Galois groups of finitely-ramified abelian extensions of $E$ become fg modules over a completion of the group ring of $Gal(E/K)$ known as the Iwasawa algebra, and one can recover information about finite subextensions from this).

More generally, given any Galois extension $E/K$ of fields, just about anything attached to $E$ -- e.g. its ring of integers, unit group, class group, etc -- will pick up a $G = Gal(E/K)$-action, and there are lots of interesting questions in this area; for instance, what does the ring of integers $\mathcal{O}_E$ look like as a module over the group ring $\mathcal{O}_K[G]$? So the Galois module structure of arithmetic objects is a very active and interesting area of research.

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As David says, the action on $Gal(L/E)$ matches up with the action on the corresponding ray class group. As far as determining the Galois group is concerned, this idea can be pushed as far is can possibly be pushable: if your total Galois group is a semi-direct product, then you can determine it completely by knowing how the quotient acts on the normal subgroup. Also, you can construct non-abelian extensions using class field theory by picking a ray class group of $E$ with the right action of $Gal(E/K)$. This is carried out in detail e.g. in section 3.1 of this paper of mine in the case of dihedral extensions. There are also lots of other papers that do similar things, many of them by Jensen, Yui, and collaborators.

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Could you clarify the aspect of a semi-direct product and how one can "determine it completely by knowin how the quotient acts on the normal subgroup"- or provide me with a reference?? I'm currently working on that in this thread. Thank you very much! – BIS HD Oct 21 '13 at 16:53

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