TITLE: Artin map restricted to base field
QUESTION [5 upvotes]: Let $M/L/K$ be a tower of local fields such that $M/L$ is abelian with Galois group $G$. The Artin map $\psi_{M/L}$ restricted to $K^\times$ is a continuous map to $G$ and thus corresponds to some abelian extension $T/K$ with an embedding $\operatorname{Gal}(T/K) \hookrightarrow G$. Challenge: Find $T$. (I am primarily interested in the mixed characteristic case, i.e. $K$ extends $\mathbb{Z}_p.$)
I conjecture the following Galois-theoretic description. Let $\tilde M$ be the normal closure of $M$. Then there is an embedding $G_1 = \operatorname{Gal}(\tilde M/K) \hookrightarrow S_n \wr G$ (where $n = [L:K]$). Let $H$ be the intersection of $G_1$ with the subgroup
$$
  \{(\sigma,g_1,\ldots,g_n) \in S_n \wr G : \sum g_i = 0 \}.
$$
Then $T = \tilde M^H$ is the $T$ we seek, and $\operatorname{Gal}(T/K) \cong G_1/H$ has a natural embedding into $G$.
Using Kummer theory, I was able to prove this when $\mu_m \subseteq K$, where $m$ is the exponent of $G$, or when $m$ is squarefree. Also the global analogue, where $M/L/K$ is a tower of number fields and we restrict $\psi_{M/L}$ to ideals of $K$, yields readily to manipulation of Frobenius elements. But there we are aided by having to consider only ideals composed of unramified primes. Indeed, the local unramified case is not hard.

REPLY [2 votes]: I've finally found the answer to my question by perusing Serre's Local Fields, Ch.XIII, specifically Propositions 10-12, which contain the functorial properties of the Artin symbol used below.
We describe $\left.\phi_{M/L}\right|_K$ in terms of $\phi_{\tilde M / K}$:
\begin{alignat*}{5}
  \text{Gal}(\tilde M / K)^{\text{ab}} &\xrightarrow{\text{Ver}}
  \text{Gal}(\tilde M / L)^{\text{ab}} &&\to \text{Gal}(M/L) \\
  \phi_{\tilde M / K}(x) & \longmapsto \phi_{\tilde M / L}(x) && \mapsto \phi_{M/L}(x).
\end{alignat*}
But these maps between Galois groups can be described as the suitable restrictions of the maps of abstract groups
$$
  (G \wr S_n)^{\text{ab}} \xrightarrow{\text{Ver}} \left(G \times (G \wr S_{n-1})\right)^{\text{ab}} \xrightarrow{\pi_1} G.
$$
Thus $T$ arises from a certain map $G \wr S_n \to G$ which, upon computation, is none other than
$$
  (g_1,\ldots,g_n,\sigma) \mapsto \sum g_i
$$
as desired.