Suppose that $K \subset L$ is a totally abelian ramified extension of local fields. Let $\pi_L$ be a prime element of $L^*.$ $F \in Gal(\tilde{L}/L)$ is the Frobenius, where $\tilde{L}$ is the maximal unramified extension of $L.$ Recall that we can define a map $$r_{L|K} : Gal(L|K) \rightarrow K^* /N_{L|K}(L^*)$$ which, when going to the abelianization of $Gal(L|K)$ defines an isomorphism. This is the Neukirch map. For more on this map see Definition 2.2 here.
Suppose now that we're in the following situation:
For $\sigma \in G(L|K)$ we have that $y^{F-1}=\pi_L^{\sigma-1}$ with $y \in U(\tilde{L})$ (here this means that $y$ is a unit of $\tilde{L},$ the maximal unramified extension of $L$, i.e it has valuation $0$).
It is known that in this case that if $K \subset L$ is a totally ramified abelian extension, then $r_{L|K}(\sigma)= N_{\tilde{L}|\tilde{K}}(y).$ The only proofs of this I know uses the norm residue symbol, see for example Serre "Local fields" XIII.5 , "Dwork's theorem."
The proof using the norm residue symbol is of course fine, but say that I didn't know that the norm residue symbol existed, how could I still prove it? Let me be more precise. Say that I knew that for any unramified extension $M \subset N$ that:
1. $M^* = N_{N|M}(N^*),$
2. If $x \in N^*$ is such that $N_{N|M}(x)=1,$ then there is a $z$ such that $x = z^{\psi-1}$ for some $\psi \in G(N|M).$
Given this, how could I deduce Dwork's theorem?
The reason I'm asking is that I've seen it stated, in Neukirch's set-up of class field theory for general profinite groups that satisfy certain conditions. This is coming from an exercise in his book "Algebraic Number Theory" before he defines the norm residue symbol, and hence my curiosity. I've tried translate the proof found in say Serre without using the norm residue symbol, but I run into problems quite early. I hope this question is appropriate for this site.
