I need some clarification in the second last paragraph from Silverman's Arithmetic of Elliptic Curves. I am not sure what 'For each $v \in M_K$ we fix an extension of $v$ to $\bar{K}$, which serves to fix an embedding $\bar{K} \subset \bar{K_v}$ and a decomposition group $G_v \subset G_{\bar{K} \ K}$.'
From what I understand, we start by considering a place $v \in M_K$, it can be extended to $\bar{K}$ in various ways, we pick the one which has the property that it fixes the above embedding and $G_v$, but I don't get how we can guarantee the existence of such a place $v$?
Could someone please break down this statement for me? I'll appreciate any help given.
Thank you.