Let $K$ be a local field, for example the $p$-adic numbers. In Neukirch's book "Algebraic number theory", there is the statement: if $K$ contains the $n$-th roots of unity and if the characteristic of $K$ does not divide $n$, and we set $L=K(\sqrt[n]{K^{\times}})$, then one has $N_{L/K}(L^{\times})=K^{\times n}$.

My questions are the following, for a (finite) Galois extension $L$ of $K$:\

(1) What happens if the characteristic of $K$ divides $n$? Can one obtain an explicit form of the image of the norm of $L^\times$?\

(2) If we don't add the condition that $K$ contains the $n$-th roots of unity, what is the image of the norm operator? If $L = K(x)$, can one write it in terms of the primitive element $x$ and $K^{\times n}$?\