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I'm referring to this old question here:

Group actions in towers of Galois extensions

I want to make this much more explicit, so I'll leave the original question below if someone answers that while I'm typing this. So let's look at $\mathbb{Q}_3$ and say we want to construct a degree $8$ Galois extension. I specifically want to use the following method, since I'm interested in this Galois action and how it can be used. We have the canonical decompositions:

$$\mathbb{Q}_3^\times \cong \mathbb{Z}\times\mu_2\times (1+3\mathbb{Z}_3),\;\;(\mathbb{Q}_3^\times)^2 \cong 2\mathbb{Z}\times 1\times (1+3\mathbb{Z}_3)$$

so $\mathbb{Q}_3^\times/(\mathbb{Q}_3^\times)^2\cong (\mathbb{Z}/2\mathbb{Z})^2$. Pick a nontrivial element $\alpha$ in this quotient group, so that quotioning with $\langle \alpha\rangle$ gives us a quadratic extension $K/\mathbb{Q}_3$ by local class field theory.

$K^\times/(K^\times)^4$ will give all the cyclic degree $4$ extensions. What we are left with is determining when such a tower gives a Galois extension. If $L/K$ is cyclic of degree $4$, then $L/\mathbb{Q}_3$ is Galois whenever an extension of an element of the Galois group of $K/\mathbb{Q}_3$ to $L$ fixes $L$.

$\mathbb{Q}_3^\times/(\mathbb{Q}_3^\times)^2$ has a canonical action on $K^\times/(K^\times)^4$, so how are the Galois actions and the actions of these groups related? Is there a nice way to translate the action of the Galois groups to simple computations in the groups of units in the corresponding fields?

I was wondering how this relates to the groups given by class field theory? In other words how can this be made explicit. Say we take $\mathbb{Q}_p$ and we have some tower of extensions


where $K/\mathbb{Q}_p$ is say cyclic of degree $2$ and $L/K$ cyclic of degree $3$. Now $K$ would correspond to some element of $\mathbb{Q}^\times/(\mathbb{Q}^\times)^2$ and $L$ would correspond to some element of $K^\times/(K^\times)^3$. Since $\mathbb{Q}^\times\subset K^\times$, I was wondering if the action of the Galois group of $K/\mathbb{Q}_p$ on the Galois group of $L/\mathbb{Q}_p$ may be expressed somehow by just using the arithmetic of $K^\times$ and $\mathbb{Q}_p^\times$?

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If I understand your question correctly, then Theorem 3.2 here should answer it completely. This is standard material, it was just easiest for me to link to my own paper. – Alex B. Jan 15 '12 at 14:28

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