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.



Artin map restricted to base field
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.


