About Local fields Let $\widehat{L}/\widehat{K}$ be an extension of local field  we know there are a number field $L$ and a place $\frak P$ of $L$ such that $\widehat{L}=L_{\frak P}.$
1) How we can prove that $\widehat{L}=\widehat{K}.L$ (in fact we must show that $L$ dense in $\widehat{K}.L$ ) ?
Denote  $G=\mathrm{Gal}(\widehat{L}/\widehat{K})$ 
2) why  $\sigma\longrightarrow\sigma_{|L}$  is an injective homomorphism of $G$ into the automorphism group of $L$? 
thank you for your help.
 A: I'm assuming that $L/K$ is finite dimensional.  It's false otherwise.  For example
if $L$ is the algebraic closure of $K=Q$, then the completion of $L$ is the field
usually called $C_p$.  But $C_p$ is definitely not $Q_pL$.
The field $L$ has two norms of interest in this situation. The first is the $p$-adic norm coming from the place. The second is the "sup"-norm on $L$ obtained by choosing a basis
for $L$ over $K$ and defining $|x|$ to be the maximum of the $K$-norms of the coefficients of $x$ in this basis.   
Now you need the theorem which says that any two norms on a finite dimensional vector space over a locally compact field are equivalent (discussed here for example.)  From this theorem you conclude that the completion of $L$ in the $p$-adic
topology is the same as the completion of $L$ with $|\cdot|$.  But the completion with $|\cdot|$ is $\hat{K}L$ pretty much by definition. 
For the second one, the point is that the Galois group $Gal(\hat{L}/\hat{K})$ acts through continuous maps, so if $\sigma_L$ is
trivial, it means that $\sigma$ fixes $K$; and since $\sigma$ is continuous, it fixes $\hat{K}$, and is thus trivial.
