I'm currently reading Serre's "Local Fields" and I'm having trouble with the exercise of Chapter 1, §6. The setup is as follows:
Let $(A, \mathfrak{m})$ be a local ring with residue field $k$, $n$ a positive integer and $f \in A[X]$ a monic polynomial of degree $n$. Let $B_f := A[X]/(f)$ and $\bar{f} = \prod_{i=1}^r \phi_i^{e_i}$ be the decomposition of $\bar{f}$ into irreducible factors in $k[X]$. For each $i$ choose a polynomial $g_i \in A[X]$ such that $\bar{g_i} = \phi_i$. Then the ideals $\mathfrak{m}_i = \mathfrak{m} + (g_i)$ are maximal in $B_f$ (here $g_i$ denotes the canonical image of $g_i$ in $B_f$). Let $C = (B_f)_{\mathfrak{m}_i}$ for some $i = 1,\dots , r$.
$\textbf{Exercise:}$ If $e_i = 1$, then the localization $C$ is a DVR.
By a proposition from the book, it suffices to show that $C$ is noetherian (and local, but this is obvious) and its maximal ideal is generated by a non-nilpotent element
I have tried proving this, however I have trouble showing that $C$ is noetherian. Any hints are greatly appreciated!
Under this generality, I don't think the statement can be true. Let $A = k[Y_1,Y_2]_{(Y_1,Y_2)}$, for a field $k$ not of characteristic 2, with $\mathfrak{m} = (Y_1,Y_2)\cdot A$ and residue field $A/\mathfrak{m} \cong k$. Let $f = X^2-Y_1-1$, and take $g_1 = X-1$ and $g_2 = X+1$. Then $e_1 = 1 = e_2$, and $B_f = A[X]/(X^2-Y_1-1) \cong A[\sqrt{Y_1+1}]$. Put $i = 1$. Then $\mathfrak{m}_1 = (X-1,Y_1,Y_2)\cdot B_f$, and the resulting localization is $C = (B_f)_{\mathfrak{m}_1}$.
Now $C /(Y_2\cdot C) \cong k[X,Y_1]_{(X-1,Y_1)}/(X^2-Y_1-1) \cong k[X]_{(X-1)}$, a domain which is not a field. So $Y_2\cdot C$ is a non-zero prime ideal of $C$ which is not maximal, hence $C$ cannot be a DVR.
Taking $A = k[Y_1,Y_2,\cdots]_{(Y_1,Y_2,\cdots)}$ and the same $f$, it is clear that non-noetherian rings $C$ can arise this way.