Definition Let $G_K$ be the absolute Galois group of a local field $K$. We will call a group homomorphism $\chi: G_K \to \mathbb{C}^*$ with finite image a character on $K$.
Since every finite subgroup of $\mathbb{C}^*$ is cyclic, it is generated by a primitive root of unity. So in our case, every character $\chi$ corresponds to a unique cyclic Galois extension $F/K$ of degree $n$, the cardinality of the image of $\chi$, and an isomorphism $$\bar{\chi}: \operatorname{Gal}(F/K) \xrightarrow{\sim} \langle \xi_n \rangle \subseteq \mathbb{C}^*$$ where $\xi_n$ is a primitive $n$-th root of unity.
Let $\chi$ be a character which is induced by a cyclic Galois extension $F/K$ with prime power degree. Furthermore, let $\operatorname{Frob}_K$ be a Frobenius element in $G_K$, i.e. an element whose image is $x \mapsto x^{|\kappa(K)|}$ under the restriction homomorphism $G_K \to G_{\kappa(K)}$ where $\kappa(K)$ is the residue field of $K$. By $F^{nr}$, we denote the maximal unramified subextension of $F/K$.
I want to show that the following statements are equivalent:
- $F/K$ is totally ramified,
- $F^{nr} = K$,
- the image of $\operatorname{Frob}_K$ in $\operatorname{Gal}(F/K)$ is the identity element,
- $\chi(\operatorname{Frob}_K)=1$.
I think I was able to show the equivalences 1. $\Leftrightarrow$ 2. and 3. $\Leftrightarrow$ 4., so I am interested in the characterization from 1./2. to 3./4 and vice versa.
Ideas:
- $\bar{\chi}$ is injective on $\operatorname{Gal}(F/K)$.
- The image of $\operatorname{Frob}_K$ under $G_K \to \operatorname{Gal}(F/K) \simeq G_K/\operatorname{Gal}(\bar{K}/F)$ is the Frobenius element in $\operatorname{Gal}(F/K)$ (Usually, a Frobenius element is unique up to conjugacy. But since $F/K$ is cyclic, it is unique indeed.). It should be the generator of $\operatorname{Gal}(F/K)$.
- The inertia subgroup $I_{F/K}$ of $\operatorname{Gal}(F/K)$ is the unique cyclic subgroup of order $e$, the ramification index of $F/K$.
- We can identify $\operatorname{Gal}(F/K)/I_{F/K}$ with $\operatorname{Gal}(F^{nr}/K)$. A generator of this group is the image of $\operatorname{Frob}_K$, I think.
Could you please help me to establish the remaining connections? Thank you!
Edit: I found a crucial mistake! $F/K$ must have prime power degree, otherwise the equivalences won't work!