Let $K(\theta)/K$ be a field extension with a minimal polynomial $f(X)$ that decompose to $f_1(X) \dots f_g(X)$ in the completion field $K_\mathfrak{p}$ for some prime $\mathfrak{p}$ of $K$.
I Know that there are exactly $\mathfrak{P}_1, \dots ,\mathfrak{P}_g$ extentions of $\mathfrak{p}$ in $K(\theta)$ and their decomposition groups fulfills:
$ D(\mathfrak{P}_i / \mathfrak{p}) \cong Gal(L_{\mathfrak{P}_i}/K_\mathfrak{p}) \cong Gal(K_p[X]/(f_i(X))) $
My question is:
For a splitting field $L/K$ of some polynomial $f(X) \in K[X]$ and $\mathfrak{P}_L/\mathfrak{p}_K$ primes, is there any relation between the minimal polynomial of the extension $L_{\mathfrak{P}}/K_\mathfrak{p}$ and $f(X)$ ?
Also, is it true to say that $ D(\mathfrak{P} / \mathfrak{p}) \cong Gal(L_{\mathfrak{P}}/K_\mathfrak{p}) \cong Gal(K_p[X]/(f(X))) $ in this case?