For a local field $E$, denote by $U(E)$ the units of the corresponding valuation ring $\mathcal{O}_E$, and denote by $U_n(E)$ the prinicipal $n$-units, i.e. $U_n(E)=1+M_E^n$ where $M_E$ is the maximal ideal of $\mathcal{O}_E$.

Now let $L/K$ be a finite unramified extension of local fields, of characteristic $0$, say. Then we have the norm map $N_{L/K}:U(L)\rightarrow U(K)$.

**The Question:** is there a simple description of $N_{L/K}^{-1}(U_n(K))$?

I know it has to contain $U_n(L)$, and by Local Class Field Theory, I know that $[U(L):N_{L/K}^{-1}(U_n(K))]=(q-1)q^{n-1}$, where $q$ is the size of the residue field of $K$. It is also not hard to find which roots of unity of $L$ are contained in $N_{L/K}^{-1}(U_n(K))$. Apart from that, I haven't been able to say much. I thought that maybe there is some simple description of this group, given that this is the unramified case, but couldn't find a reference.

Thank you!