roots of unity, wild ramification and units of norm one in local fields

Let $\mathbf{K}$ be a finite Galois extension of $\mathbf{Q}_p$ containing the $p$-th roots of unity and let $\mathfrak{p}$ denote the unique prime of $\mathbf{K}$ lying above $p$. For each $n \geq 1$ we consider the extension $\mathbf{K}_n$=$\mathbf{K}(\zeta_{p^{n+1}})$, where $\zeta_{p^{n}}$ denotes the $p^n$-th root of unity and the following objects attached to $\mathbf{K}_n$:

• $\mathfrak{p}_n$ is the unique prime in $\mathbf{K}_n$ lying above $p$;
• $U_1(\mathbf{K}_n)$ is the group of units in $\mathbf{K}_n$ which are 1 modulo $\mathfrak{p}_n$;
• $U_n'$ is the subgroup of $U_1(\mathbf{K}{_n})$ consisting of those units whose norm to $\mathbf{Q}_p$ is equal to 1.

I am trying to establish whether the norm map $N_{\mathbf{K}_n/\mathbf{K}_m}:U_n' \to U_m'$ is surjective for all $n \geq m$. In order to solve this using class field theory, it seems to me that it is relevant to establish first whether the extensions $\mathbf{K}_n/\mathbf{K}$ are wildly ramified or not. So another (elementary) question is whether $\mathbf{K}_n/\mathbf{K}$ is indeed wildly ramified or not. This post (https://mathoverflow.net/questions/136052/cyclotomic-extension-of-p-adic-fields) shows that more caution than in the case of $\mathbf{Q}_p(\zeta_{p^{n+1}})/\mathbf{Q}_p(\zeta_p)$ is required, but one of the comments also says that $\mathbf{K}_n/\mathbf{K}$ is wildly ramified for all $n$ large enough. Why is this case?

• So I guess that the failure of total wild ramification comes from some unramified behavior creeping in? Always? Do you have examples beyond those given by Chandan Singh Dalawat? Sep 7, 2017 at 18:06
• No, that is the only reference I found regarding wild ramification. Would the result be true if we knew the wild ramification bit? Sep 7, 2017 at 20:34
• I ought to have the tools to answer this, but I’m occupied with a nonmathematical enterprise with a deadline. I’ll try to slip some time in to look at the problem, but I can’t guarantee. I haven’t even looked at the question of whether the total degree over $K$ is a power of $p$ — is that clear to you? Best hope here is for a real expert to chime in. Sep 8, 2017 at 18:02
• “but one of the comments says…” — that can’t possibly true as quoted, since if there’s some unramified behavior at a low level, it doesn’t disappear when you make bigger and bigger extensions. Meanwhile, I have started looking at the problem. Maybe… Sep 9, 2017 at 4:00
• Thank you very much for this. In the meantime I have found a reference (L. Washington-Introduction to cyclotomic fields, second edition, Lemma 13.53) where the result is proved in the special case $\mathbf{K}=\mathbf{Q}_p(\zeta_p)$. I think that proof could be adapted if one knew the wild ramification part, but I am not entirely sure. Sep 9, 2017 at 9:01

1) About "wild ramification" in the cyclotomic extension $K_n/K$ (your notations). It will be convenient to introduce the cyclotomic $\mathbf Z_p$-extension $K_{cyc} = \cup K_n$, which is Galois over $K$ with group isomorphic to $(\mathbf Z_p, +)$. The inertia subgroup $I$ is necessarily of the form $p^a\mathbf Z_p$, and $a=0$ iff $K_{cyc} /K$ is totally ramified. But, in @Lubin's terms, "some unramified behavior [could be] creeping in", more precisely the intersection $K_I = K_{cyc} \cap K_{nr}$ could be larger than $K$, where $K_{nr}$ denotes the unramified $\mathbf Z_p$-extension of $K$ . In the kummerian situation here (i.e. $K$ contains $\zeta_p$) it is not hard to determine $K_I$ inductively. Notations : let $e'_K =\frac 1 {(p-1)}ord_k (1-\zeta_p)$ (this is an integer); for $x \in K^*$, define $def_K (x) =$ max $ord_K (y-1)$ for all $y \in$ the class of $x$ mod $(K^*)^p$ ("def" is for "defect"). It is easily shown that a cyclic extension of degree $p$, $L=K(\sqrt [p]x)$, is unramified iff $def_K (x) = pe'_K$ (1). If $L$ is unramified, repeat the process on replacing $K$ by $L$, and so on until you reach $K_I$.
2) Given $n$, you want to show that the norm $U'_m \to U'_n$ (your notations) is surjective for all $m \ge n$. To ease the notations, take first $K_n$ to be the base field $K$. Then the intersection $\mathcal N_K :=\cap N_{K_n/K}(K_n^*)$ is called (for obvious reasons) the subgroup of "universal (cyclotomic) norms" of $K_{cyc}/K$. Suppose for simplification that $p$ is odd (otherwise, replace $p$ by $4$).By Kummer and local CFT (2), it is known that $x\in \mathcal N_K$ iff $N_{K/\mathbf Q_p}(x) \in p^{\mathbf Z}$. Restricting to the principal units, you get that $\mathcal N_K \cap U_1 (K) = U'_K$ . This conclusion is of course valid at any level $K_n$. Finally, one can show by direct arguments of local compacity the surjectivity of the norm $\mathcal N_{K_m} \to \mathcal N_{K_n}$ for all $m\ge n$ (2). So you are done.