I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3.

# Notation for the problem

Let $G$ be a profinite $p$-group (so all quotients by open subgroups are $p$-groups). We will denote the closed subgroups of $G$ by $G_K$ and call the indices $K$ for fields. We say that $K$ is the fixed field of $G_K.$ We write $K \subset L$ if $G_L \subset G_K$ and we call $K \subset L$ a field extension. We say that it is a finite extension if $G_L$ is of finite index in $G_K$ and we say that it is Galois if $G_L$ is normal in $G_K$ and then we write $G(L|K) = G_K / G_L.$ We call this the Galois group of $K \subset L.$

Further, let us say that we have a continuous multiplicative $G$-module $A.$ We mean with this a multiplicative abelian group $A$ on which $\sigma \in G$ act as automorphisms on the right and they act continuously. We let $A_K = \{a \in A | a^\sigma = a, \forall \sigma\in G_K\}.$Clearly we then have that $A_K \subset A_L$ for any field extension $K \subset L.$ If $$K \subset L$$ is finite, then we have a norm map $$N_{L|K}:A_L \rightarrow A_K,$$ $$N_{L|K}(a) = \Pi_\sigma a^\sigma$$ where $\sigma$ varies over a system of representatives of $G_K / G_L.$

Now, suppose further that we have a surjective map $$d:G \rightarrow \mathbb{Z}_p$$ where $p$ is a prime number. We also suppose we have a map $$v:A_k \rightarrow \mathbb{Z}_p$$ which we call a **henselian $p$-valuation** with respect to $d$ and assume that $v$ satisfies the following properties:

- $$v(A_k) = Z \supset \mathbb{Z}$$ and $$Z/nZ \cong \mathbb{Z}/n\mathbb{Z}$$ for all $n$ that are a power of $p.$
- $$v(N_{K|k}(A_K))=f_k Z$$ for all finite extensions $k \subset K$ where $$f_K = (d(G):d(G_K)).$$

Suppose that $A$ satisfies the class field axiom, which means that for every cyclic extension $L|K$

$$ |H^i(G(L|K),A_L)| = \begin{cases} [L:K] & \text{for } i= 0 \\ 1 & \text{for } i=-1. \end{cases}$$
Here $$H^0(G(L|K),A_L) = A_K/N_{L|K}(A_L)$$ while $$H^{-1}(G(L|K),A_L) = \text{ ker } N_{L|K} / I_{G(L|K)}$$
where $\text{ker } N_{L|K}$ is the kernel of $N_{L|K} : A_L \rightarrow A_K$ and $I_{G(L|K)}$ is the ideal generated by elements of the form $a^{\sigma-1} = a^\sigma \cdot a^{-1}$ for $a \in A_L$ and $g \in G(L|K).$

# The problem statement

With notation now set up, the question is as follows. Suppose that we have a p-class field thoery $(d:G \rightarrow \mathbb{Z}_p,v:A_k \rightarrow \mathbb{Z}_p)$ as above. Suppose we have another surjective homomorphism $d':G \rightarrow \mathbb{Z}_p.$ Let $\text{ker } d' = G_{\hat{K'}}.$ We then have that $\hat{K'} |K$ is an extension with galois group isomorphic to $\mathbb{Z}_p.$ Let now $$v':A_k \rightarrow \mathbb{Z}_p$$ be the composite of $$A_k \xrightarrow{(,\hat{K'}|K)} G(\hat{K'} |K) \xrightarrow{d'} \mathbb{Z}_p$$ where $$(,\hat{K'}|K)$$ is the norm residue symbol derived from the $p$-class field theory.

Neukirch now claims that:

**$(d',v')$ is also a p-class field theory.**

I thought I had a proof of this, but I a now very unsure, since he states that the theorem does not hold for $\hat{\mathbb{Z}}$ class field theories, i.e if we replace $\mathbb{Z}_p$ with $\hat{\mathbb{Z}}$ above and $G$ is any profinite group, not neccesarily a $p$-group, the analogous statement does not hold. So my question is two-fold:

- How can I prove that $(d',v')$ is a p-class field theory?
- What goes wrong for $\hat{\mathbb{Z}}?$ What makes the case so different compared to $\mathbb{Z}_p?$