Let $K$ be a local field. In the standard book: Algebraic number theory, written by Jurgen Neukirch, there is one statement:

If $K$ contains the $n$-th roots of unity and if the characteric of $K$ does not divide $n$, we consider the extension $L=L(\sqrt[n]{K^{\times}})$ over K, then one has $N_{L/K}(L^{\times})=K^{\times n}$.

My questions are following:

Assume L/F is galois field extension :

(1) What happens if characteric of $K$ divide $n$ , can one obtain the explicit form of the image of the norm ?\

(2) If we don't add the condition: $K$ contains $n$-th roots, what is the image of the norm operator, it may be relative to $K^{\times n}$, can one write it in terms of the generator $x$ and $K^{\times n}$ if we know L=F(x) ?\