From Silverman's AEC page 332: 


I need to understand why the determination of the following local kernel 
$$ 
ker \Big( H^1(G\_v, E[\phi]) \rightarrow WC(E/K\_v)[\phi] \Big)
$$


is straightforward. The book says that it is the same as answering the questions whether a curve has a point over a complete local field (which I understand). 


**This is further reduced by Hensel's Lemma to checking whether the curve has a point in some finite ring $R\_v/ \mathcal{M}\_v^e$ for some easily computable integer $e$.**


Now, as far as I know and understand, Hensel's Lemma says that for a polynomial $f(x) \in R\_v[x]$, if it has a root in $R\_v/\mathcal{M}$, then it lifts to a unique root in $R\_v$ and hence $K\_v$. 
However, this version does not seem to be directly used here. I suspect that maybe some several variable version is being used? And where did that $e$ come from? 


I would really appreciate it if someone could explain it to me. Thank you.



Clarification: Using Hensel's Lemma to determine $K_v$-rational points on a curve
you really shouldn't crosspost. Anyway, you've slightly misstated Hensel's lemma, you left out the assumption that $f(x)$ has a **simple** root in $R\_v/\mathcal{M}\_v$. That's where the $e$ is coming from. In general, if $f(x)$ has a root of higher multiplicity in $R\_v/\mathcal{M}\_v$, then you need to work in $R\_v/\mathcal{M}\_v^e$. So if the curve is non-singular modulo $v$, then you can take $e=1$, and it really does reduce to the 1-variable Hensel lemma, since the curve has dimension 1. However, if the curve is singular and it has a singular point defined over $R\_v/\mathcal{M}\_v$, that's not enough to conclude that the point can be lifted, you need to work with a larger value of $e$.


