I'm trying to find a proof for the following statement:
Let $K_\mathfrak{p}$ be a complete field such that $\bar{K_\mathfrak{p}}$ is algebraically closed , let $f(x) \in K_\mathfrak{p}[X]$ be a polynomial. Now let's assume that the reduction of $f$ to $\bar{K_\mathfrak{p}}$ has the form $(X-\bar\alpha)^e$. So the field extension $L = K_\mathfrak{p}/f\cong K_\mathfrak{p}[X]/(f(x))$ is galois