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 $K^{\times n}$?\



local class field theory (Norm map)
Let $L/K$ be a finite abelian extension of local fields. Although, there is no generic form for the image of the norm map, $N^{L}\_K$, in practice one can follow the following procedure to determine its image. 


Choose a uniformizer $\pi\_L$ in $\mathcal{O}\_L.$ Then $L^{\times}$ is equal to the group generated by $\pi\_L$ and $\mathcal{O}\_L^{\times}.$ It follows that $N^{L}\_K L^{\times} = \langle N^{L}\_K(\pi\_L) \rangle N^{L}\_K \mathcal{O}\_L^{\times},$ Hence to determine $N^{L}\_K L^{\times}$ it is enough to establish the image of $\mathcal{O}\_L^{\times}$ under the norm mapping. 


The group $N^L\_K \mathcal{O}\_L^{\times}$ is a subgroup of $\mathcal{O}\_K^{\times}$ and by a group cohomology argument it can be shown that


$[\mathcal{O}\_K^{\times}:N^{L}\_K\mathcal{O}\_L^{\times}] = e(L|K) = [L^{\times} : K^{\times}\mathcal{O}\_L^{\times}].$


In particular, if $L/K$ is unramified, $\mathcal{O}\_K^{\times} =N^{L}\_K\mathcal{O}\_L^{\times}$ and hence $N^{L}\_K L^{\times} = \langle N^{L}\_K(\pi\_L) \rangle \mathcal{O}\_K^{\times}.$


The norm group of a tamely ramified extension is similarly easy to deduce. Write 


$$\mathcal{O}\_L^{\times} = \langle \zeta\_{q\_L - 1} \rangle U\_L$$ 


where $q\_L$ is the residue field characteristic of $L$ and $\zeta\_n$ denotes a primitive n-th root of unity. Denote the residue field of $L$ by $l$ and that of $K$ by $k,$ then 


$$N^L\_K(\zeta\_{q\_L -1}) = N^l\_k(\zeta\_{q\_L -1})^{e(L|K)} = \zeta\_{q\_K -1}^{e(L|K)}.$$


As $U\_K$ is a pro-$p$ group and contains the image of $U\_L$ under $N^L\_K,$ we have in the tamely ramified case that $U\_K = N^L\_K(U\_L)$ else $\mathcal{O}\_K^{\times}/N^{L}\_K\mathcal{O}\_L^{\times}$ would contain an element of $p$-power order contradicting the equality 


$$[\mathcal{O}\_K^{\times}:N^{L}\_K\mathcal{O}\_L^{\times}] = e(L|K)$$


and the fact that $L/K$ was assumed tamely ramfied. It follows in the tamely ramified case that $$N^{L}\_K L^{\times} = \langle N^{L}\_K(\pi\_L), \zeta\_{q\_K -1}^{e(L|K)}\rangle U\_K$$


The case of wild ramification is more difficult. But two facts are helpful. First, in the case $K$ is a p-adic field $U\_K^{ap^n} \supset 1 +\mathcal{M}\_K^{2ne(k|\mathbb{Q}\_p) + 1}$ where $a$ and $p$ are relatively prime. Hence, it is enough to determine the image of the norm mapping in the units of $\mathcal{O}\_K \mod \mathcal{M}\_K^{2ne(k|\mathbb{Q}\_p) + 1},$ a finite set. 


Another technique is to determine the higher ramification filtration of $Gal(L/K)^{ab}.$ In practice this can be done by examining the derivatives of the irreducible polynomial of $\pi\_L.$ These groups map under the inverse of the artin map to the higher unit groups. As the domain of the inverse of the artin map is $K^{\times}/N^{L}\_K L^{\times}$ their sizes reveal norm indexes within the higher unit groups. For a good exposition see Serre's book Local Fields. 


