In Chapter I, Section 6 (Simple Extensions) of Serre's Local Fields, we see the following proposition and pair of corollaries:
Assumptions: $A$ a discrete valuation ring, $B_f = A[X]/(f)$ for some polynomial $f$. $\mathfrak{m}$ is the unique maximal ideal of $A$ with residue field $k=A/\mathfrak{m}$, and $\overline{f}$ the image of $f$ in $k[X]$. $K$ is the field of fractions of $A$.
Proposition 15: If $\overline{f}$ is irreducible, then $B_f$ is a discrete valuation ring with maximal ideal $\mathfrak{m}B_f$ and residue field $k[X]/(\overline{f})$.
Corollary 1: If $\overline{f}$ is irreducible, $f$ is irreducible in $K[X]$. Letting $L=K[X]/(f)$, the ring $B_f$ is the integral closure of $A$ in $L$.
Corollary 2: If $\overline{f}$ is a separable polynomial, the extension $L/K$ is unramified.
My Question: Why does Corollary 2 assume separability? And what definition of unramified extension is Serre using? Firstly, I don't know if Serre is assuming $\overline{f}$ is irreducible for Corollary 2, but it seems that if he is not, the field $L$ is not well-defined. Assuming $\overline{f}$ is irreducible and separable, letting $\ell$ be residue field of $L$, we should have $[L:K]=[\ell:k]$ (since $f$, $\overline{f}$ are irreducible of the same degree), and so the extension is unramified, with no discussion of separability necessary.
