# Why is a general element in $G(L/K)$ a power of the Frobenius automorphism when restricted to $L \cap \tilde{K}$?

I'm going through Neukirch's Algebraic Number theory and I'm stuck on why we can say that a general element in $$G(L/K)$$ is a power of the Frobenius automorphism when restricted to $$L \cap \tilde{K}$$ in the proof of proposition 4.4.

Prop. : Given a finite Galois extension $$L/K$$, then the mapping $$\{\sigma \in G(\tilde{L}//K) | d_{K} (\sigma)=n \in \mathbb{N}^{+} \} \rightarrow G(L/K)$$ is surjective.

In the proof it is just stated that given $$\sigma \in G(L/K)$$ when restricted to $$L \cap \tilde{K}$$ it is a power of the Frobenius automorphism restricted to $$L \cap \tilde{K}$$. And I can't think of a reason why it should be true.

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• Please type up the relevant part of the proposition. Commented Jul 17, 2020 at 16:50

This is Chapter IV of Neukirch, where he works with abstract Galois theory. Here is an argument that any $$\sigma \in G(L|K)$$ is a power of the Frobenius in that setting, rather than the special case where $$G$$ is the Galois group of local fields.
In abstract Galois theory $$G_K$$, $$G_L$$ are closed subgroups of a profinite group $$G$$, and $$G(L|K) = G_K/G_L$$. Also there is a surjective map $$d: G \rightarrow \widehat{\mathbb{Z}}$$, $$f_K = (\widehat{\mathbb{Z}} : d(G_K))$$, $$d_K = (1/f_K)d$$, the Frobenius $$\varphi_K$$ is defined via $$d_K(\varphi_K) = 1$$, and he writes $$\varphi_{L|K}$$ for $$\varphi_K \bmod G_L$$. Unramified means $$(d(G_K):d(G_L)) = (G_K : G_L)$$.
It is enough to show that if $$L$$ is unramified over $$K$$ then $$G(L|K) = ( \varphi_{L|K} )$$, that is, $$G(L|K)$$ is the cyclic group generated by $$\varphi_{L|K}$$. For this, suppose $$|G_K/G_L| = n$$, then $$$$n = (G_K : G_L) = (d(G_K) : d(G_L)) = (d_K(G_K) : d_K(G_L))$$$$ Since $$d_K(G_K) = \widehat{\mathbb{Z}}$$, $$d_K(G_L) = n\widehat{\mathbb{Z}}$$ (using facts about $$\widehat{\mathbb{Z}}$$ presented in Example 4 on page 272). The $$\{\varphi_{L|K}^j \}_{j=0}^{n-1}$$ are distinct mod $$G_L$$, because $$d_K(\varphi_{L|K}^j) = j$$ and so $$\varphi_{L|K}^j$$ cannot be in $$G_L$$ since $$j = d_K(\varphi_{L|K}^j) \notin d_K(G_L) = n\widehat{\mathbb{Z}}$$ for $$j < n$$. So the $$n$$ powers $$\{\varphi_{L|K}^j\}_{j=0}^{n-1}$$ account for all of $$G_K/G_L$$.