Completion of Galois extension is Galois

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.

• 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 $L_w$. Sep 24 '21 at 22:13