Relating Galois groups related via completions

Question

Let $K$ be a number field, and let $K_\mathfrak p$ denote the completion of $K$ at the prime $\mathfrak p$ of $K$. I'm wondering what can be said (if anything useful) that relates the Galois groups $\mathrm{Gal}(\bar{K}/K)$ and $\mathrm{Gal}(\bar{K_\mathfrak p}/K_\mathfrak p)$? If anyone knows relevant textbooks etc. that would be great too.

Answer 1

The local Galois group is isomorphic to a closed subgroup of $G_K=\mathrm{Gal}(\overline{K}/K)$, but this subgroup is only determined up to conjugacy. If you fix an embedding $\overline{K}\hookrightarrow\overline{K}_\mathfrak{p}$ over $K\hookrightarrow K_\mathfrak{p}$, then you get a corresponding continuous homomorphism $G_{K_\mathfrak{p}}\to G_K$ which is injective by Krasner's lemma. If you change the embedding of fields, you change the embedding of Galois groups by a conjugation in $G_K$. The image of the map $G_{K_\mathfrak{p}}\hookrightarrow G_K$ is the decomposition group in $G_K$ of the prime $\mathfrak{p}_\infty$ of $\overline{K}$ lying over $\mathfrak{p}$ induced by the choice of embedding $\overline{K}\hookrightarrow\overline{K}_\mathfrak{p}$.