Let $K$ be a global number/function field, and let $v$ be a place of $K$. How to construct an explicit map from $G(\overline{K}/K)\rightarrow G(\overline{K}_{v}/K_v)$?
Map from absolute Galois group of a global field to that of one of its completions
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galois-theory
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1Indeed; if $L$ is the algebraic closure of $K$ in $K_v$ then $G(\overline{K}_v,K_v)\cong G(\overline{K}/L)\subset G(\overline{K}/K)$. For accepting, see e.g. http://math.stackexchange.com/faq#howtoask (click on your name to see the list of the questions you asked so far) – 2012-03-07