Let K be a p-adic field, L a totally ramified finite extension of K. Prove that $N_{L/K}(U^1_L)=U^1_K$ iff L/K is tamely ramified, where $U_K^1=1+\pi_K\mathcal{O}_K,U_L^1=1+\pi_L\mathcal{O}_L$ are the 1st higher unit groups.
I believe this problem has something to do with class field theory, but I had little grasp on that subject... Any help would be appreciated.
Since it seems that you may like it, I'll expand on reuns's comment.
We need only show the equality stated in the comment. Take $\lambda \in O_{K}$, we want to find $x\in O_{K}$ such that $$(1+\pi_{K}x)^{n}=1+\pi_{K}\lambda$$ is satisfied.
By looking at the development of the left hand side, this is equivalent to find an $x\in O_{K}$ such that the equation $$\lambda=x(n+x\pi_{K}\binom{n}{2}+...+x^{n-1}\pi^{n-1})$$ is satisfied.
But by going to the residue field this becomes: $[\lambda]=[n]x$, which has always a solution in $O_{K}$ thanks to Hensel's lemma which we can apply because $p$ does not divide $n$.