Let $L/K$ be a finite Galois extension, $v$ be a non-arquimedian valuation of $K$ and $w$ an extension of $v$ to $L$. Denote by $L_w$ and $K_v$ the completions. Neukirch's Algebraic Number Theory states without prove that $L_w/K_v$ is a finite Galois extension. To justify this statement properly, I tried to proceed like this:

Denote by $G_w$ the decomposition group of $w\mid v$. Every $\sigma\in G_w$ is continuous with respect to $w$, so it admits an extension to an $K_v$-automorphism $\hat{\sigma}$ of $G_w$. This induces an injective homomorfism $\sigma\mapsto\hat{\sigma}$. Since $[L_w:K_v]=ef$, I would prove that $[L_w:K_v]$ is Galois if I could prove that $| G_w|=ef$. If $v$ is discrete, then it follows from the fundamental identity. However, I don't know how to prove it in the general case.

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    $\begingroup$ I don't think you need the Decomposition group to prove that the extension is Galois. Write $L = K(\alpha)$. Then show that $L_w = K_v(\alpha)$. Since all the roots of $\alpha$ are contained in $L$ (by assumption), the same is true for the larger extension $L_w$. $\endgroup$
    – Mathmo123
    Sep 24 '21 at 22:13

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