Neukirch's class field axiom and cohomology of units for unramified extension

Question

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.

Answer 1

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 $\pi_K$.