The following is an exercise in Serre's book on Local Fields:
$A$ is a local ring with residue field $k = A/\mathfrak{m}$. Let $f\in A[x]$. Then, if $\overline f$ factors as $\overline f = \prod_{i\in I}\phi_i^{e_i}$ in $k[x]$, then let $g_i\in A[x]$ be such that $\overline{g_i} = \phi_i$.
Question: Show that if $e_i = 1$, then $\left(A[x]/(f)\right)_{(\mathfrak{m},g_i)}$ is a DVR.
I at first mistakenly thought that since $A$ is a local ring, we could say that $\left(A[x]/(f)\right)_{(\mathfrak{m},g_i)}= (A[x]/(f))_{(g_i)}$, but then if, say, $g_i = f = x-1$ you'd get $(A[x]/(x-1))_{(x-1)} = k$, which is not a DVR.
I then tried to use literal definitions of localization, which haven't helped in the slightest. I feel this shouldn't be as hard as I'm making it, and any hints would be appreciated! Thanks in advance!