I have a question about a statement in Chapter V, section 3 of Serre’s book on local fields.
Let $\ell$ be a prime number, and let $L/K$ be a cyclic Galois extension of local fields of degree $\ell$, totally ramified. Let $t$ be the (first) break number, meaning that the ramification groups of the extension are $G_0=\ldots=G_t\simeq\mathbb Z/\ell\mathbb Z$, and $G_{t+1}=\ldots=1$. It is known that $t\ge1$.
Corollary 3 of loc.cit. makes the following statement about surjectivity of the norm map on higher unit groups: $N(U_L^{\psi(n)})=U_K^n$ for $n>t$ and $N(U_L^{\psi(n)+1})=U_K^{n+1}$ for $n\ge t$, where $\psi$ is the function in Herbrand’s theorem.
Since $t\ge1$, this statement makes no claim about the image of the norm on the principal units $U_L^1$. Of course, $N(U_L^1)\subseteq U_K^1$. Is it true that $N(U_L^1)\subsetneq U_K^1$?
The answer is given in the same book of Serre, in Chapitre XV, §2, Corollaire 2 to Théorème 1. This states the following. Let $L/K$ be a totally ramified abelian Galois extension of complete discretely valued fields with (quasi-)finite residue fields. Let $c$ be the largest integer such that the lower ramification group $G_c$ is not trivial, and let $f=\varphi(c)+1=\frac{|G_0|+|G_1|+\ldots+|G_c|}{|G_0|}$. Then $U_K^f\subseteq N_{L/K}(L^\times)$ and $U_K^{f-1}\not\subseteq N_{L/K}(L^\times)$.
In particular, if $L/K$ is totally tamely ramified, then $\mathrm{Gal}(L/K)=G_0$ and $G_1=1$, thus $c=0$ and $f=1$, so $U_K^1$ is contained in the image of the norm map. If $L/K$ is totally wildly ramified, then $\mathrm{Gal}(L/K)=G_0=G_1$, so $c\ge1$, $f\ge2$, and $U_K^1$ is not contained in the image of the norm map.