Im facing a problem in Silvermans Book "Arithmetic of elliptic Curves" at the beginning of chapter X.4 concerning the exact sequences.
Let $K$ be a number field with a valutaion $v$. I'm distinguishing between the algebraic closure of $K_v$, which i denote as $\overline{K_v}$ and the completion of the algebraic closure of $K$, which is defined as $L = \bigcup L_{iv}$ with the $L_i$ ranging over all finite extensions of $K$ (as done in Neukirch). Now to obtain the short exact sequence $(\ast)_v$ used to define the Selmer-Group in Silverman's "Arithmetic of Elliptic Curves", page 331:
$ 0 \longrightarrow E'(K_v)/\phi E(K_v) \longrightarrow H^1 (G_v, E[\phi]) \longrightarrow H^1 (G_v, E)[\phi] \longrightarrow 0 $
I have to work with $\overline{K_v}$, as I need an algebraically closed field. However, how can one let $G_v\subset Gal( \overline{K} | K)$ act on $\overline{K_v}$? I know $G_v \simeq Gal(L | K_v)$, but do not see, how to use that isomorphism here. On the other side, if I work with $L$, I don't know how to identify $H^1 (G_v, E)$ with the Weil-Chatelet-Group $WC(E/K_v)$, which is in bijection to $H^1 (Gal(\overline{K_v}|K_v), E)$.
Im very thankful for every advice given!
For those of you not having Silverman's book at your hands, it can be found here http://www.mathe2.uni-bayreuth.de/stoll/talks/short-course-descent.pdf. The section described above begins on page 331.
The two fields you are considering are the same, because of Krasner's lemma, which has as a corollary that separable closure and $p$-adic completion commute for global fields. See for example prop. 8.1.5 of Neukirch's "Cohomology of number fields".