nt.number theory - Neukirch's class field axiom and cohomology of units for unramified extension

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.)