I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.

I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and Vladimir Dokchitser:

Theorem 1Every Frobenius-semisimple Weil representation $\rho$ is uniquely determined by its local polynomials $P(\rho/F,T)$ over finite separable extensions $F/K$.

Before we talk about the cyclic case of the proof, let us recall some **definitions** first:

- Let $K$ be a local field and $G_K = \operatorname{Gal}(\bar{K}/K)$ be the absolute Galois group of $K$. An
**(arithmetic) Frobenius element**is any element $\operatorname{Frob}_K \in G_K$ that acts as $x \mapsto x^{|k|}$ on $\bar{k}$, the algebraic closure of the residue field $k$ of $K$. - The
**Weil group**$W_K$ is the subgroup of $G_K$ of all automorphisms that act as an integral power of Frobenius on the residue field. - A
**Weil representation**is a representation $\rho: W_K \to \operatorname{GL}_n(\mathbb{C})$ such that $\rho(I_K)$ is finite. It is called**Frobenius-semisimple**if the image of some (equivalently, any) Frobenius element is diagonalizable. - The
**local polynomial**$P(\rho,T)$ is the inverse characteristic polynomial of $\operatorname{Frob}_K^{-1}$ on the inertia invariants of $\rho$, i.e. $$P(\rho,T) = \det(1-T \cdot \operatorname{Frob_K^{-1}}).$$ Similarly, for a finite extension $F/K$, we write $P(\rho/F)$ for the local polynomial of the restriction of $\rho$ to $W_F$, i.e. $$P(\rho/F,T) = P(\rho|_{W_F},T).$$

Now I would like to talk about the **proof of the cyclic case** which is given in the paper:

Step 1: Cyclic.Suppose $\rho$ factors through a finite cyclic group $G = \operatorname{Gal}(F/K) \simeq C_n$ and $F/K$ has ramification degree $e$. By Lemma 2 (cf. below), there is a cyclic totally ramified extension $K/K$ of degree $e$ such that $FL/L$ is unramified of degree $n$. The restriction map $\operatorname{Gal}(FL/L) \to \operatorname{Gal}(F/K)$ is an isomorphism, as it is clearly injective and both groups have order $n$. So $\rho/L$ determines $\rho$, and $\rho/L$ can be recovered from its local polynomial $P(\rho/L,T)$.

In our proof we used the following **Lemma** which we shall take for granted in this post:

Lemma 2Let $F/K$ be a cyclic extension of degree $n$ and ramification degree $e$. Then there exists a cyclic totally ramified extension $L/K$ of degree $e$ such that $FL/L$ is unramified of degree $n$.

Now I have the following specific **questions** about the proof above:

- Could you give me a good argument why the restriction map $\operatorname{Gal}(FL/L) \to \operatorname{Gal}(F/K)$ is injective? Let us say we have $\sigma, \sigma' \in \operatorname{Gal}(FL/L)$ which are not equal. Then there exists an $x \in FL \setminus L$ such that $\sigma(x) \neq \sigma'(x)$. If $x \in F$ then the restrictions $\bar{\sigma}, \bar{\sigma}$ are obviously not equal. But what happens if $x$ is not in $F$?
- Why does $\rho/L$ determine $\rho$ then? And how can we recover $\rho/L$ from its local polynomial $P(\rho/L,T)$? What do "determining" and "recovering" even mean in these cases?

**Additional Remark**:

I would also like to add a **diagram** which my professor drew to explain me something relating to this Theorem (or Lemma?):

$\require{AMScd}\DeclareMathOperator\Gal{Gal}$ \begin{CD} @. \Gal(L F/L) @>{\text{$\chi|_L$, unramified}}>> \Gal(K^{\mathrm{ur}}/K) \\ @. @V{\simeq}VV @| \\ 1 @>>> \Gal(F/K^{\mathrm{ur}}) @>>> \Gal(F/K) @>>> \Gal(K^{\mathrm{ur}}/K) @>>> 1 \\ @. @. @V{\chi}VV \\ @. @. \mathbb C^\times \end{CD}

And next to this diagram, he also wrote $P(\chi|_L,T) = 1 - \operatorname{Frob}_L \cdot T$.

To this remark, I have the following **questions** (if it is really related to the proof):

- What is $\chi$ supposed to mean?
- Is the diagram commutative?
- Why all of the sudden does the maximal unramified extension (I think?) $K^{ur}$ of $K$ appear here?
- If or how is this related to the proof of Theorem 1 (resp. Lemma 2)?

Could you please help me answering my questions? I feel like I lack a lot of background knowledge which is required here, so it would be really nice if you could explain them to me carefully. Thank you in advance!