This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in Chapter IV, Proposition 6.2, that his class field axiom implies that the Tate cohomology groups $H^n(G(L|K),U\_L)$ for $n=0,-1$ vanish for finite unramified extensions $L|K$, where $U\_L$ is the group of units. He mentions in the proof that every element $a \in A\_L$ can be written as $a = \epsilon \pi\_K^m$, where $\epsilon \in U\_L$ and $\pi\_K$ is a prime element in $A\_K$. Why does this work?
I absolutely understand this argument when the image of the valuation just lies in $\mathbb{Z}$! But how does this work for a valuation whose image is $\hat{\mathbb{Z}}$? Unless $A$ is not a profinite module, I don't know what $\pi\_K^m$ is for some general $m \in \hat{\mathbb{Z}}$. Unfortunately, this must work in this generality for global class field theory.



Neukirch's class field axiom and cohomology of units for unramified extension
You don't need to make sense out of $\pi\_K^m$ for a general $m$ in $\hat{\mathbb{Z}}$. All you really need to know for his argument is that $v\_K(A\_K) = v\_L(A\_L)$ as subgroups of $\hat{\mathbb{Z}}$. I didn't think this through but I think it should be pretty easy to establish from the fact that $\pi\_K$ is prime for both valuations.


All he really uses is that the Galois group fixes $\pi\_K$.


