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 \ZZ! But how does this work for a valuation whose image is \widehat{\ZZ}? Unless A is not a profinite module, I don't know what \pi_{K}^m is for some general m \in \widehat{\ZZ}. Unfortunately this has to work in this generality for global class field theory.

(\ZZ denotes the integers of course, sorry for my personal notation.)