This is a question from Milne's 'Algebraic Number Theory'. Let $K$ be a valued field with absolute value $|\cdot|$ and $L=K(\alpha)$ a finite separable extension of $K$. Let $\hat{K}$ be the completion of $K$ with respect to $|\cdot|$ and $f(X)$ the minimum polynomial of $\alpha$ over $K$.
Suppose $g(X)$ is a monic, irreducible factor of $f(X)$ over $\hat{K}$ and let $x:=X+(g(X))$ so that $\hat{K}[x]=\hat{K}[X]/(g(X))$.
We have a diagram $\require{AMScd}$ \begin{CD} L @>\displaystyle \alpha\mapsto x >> \hat{K}[x]\\ @VVV @VV V\\ K @>>> \hat{K} \end{CD}
Since $\hat{K}[x]/\hat{K}$ is algebraic, the valuation on $\hat{K}$ extends uniquely to $\hat{K}[x]$.
My question: Does the above diagram commute, i.e. does this valuation on $\hat{K}[x]$ induce a valuation on $L$ which extends $|\cdot|$?
Many thanks in advance.