We are given a complete discrete valuation field $K$ and its finite extension $L$. Denote $k=\mathcal O_K/ \mathfrak{m}_K$ and $l=\mathcal O_L/ \mathfrak{m}_L$. Assume that $k$ is a perfect field. I want to proof the following criterion.
$K\subset L$ is unramified iff $L=K(\alpha),$ where $\alpha\in \mathcal O_L,~$ $p\in \mathcal O_K[x]$ is minimal polynomial for $\alpha$ and $\overline{p}\in k[x]$ is separable.
I fully understand the proof for the $\implies$ direction, but I am having some trouble with $\impliedby$ direction.
To be precise, while proving $\impliedby$, if $\overline{p}$ is guaranteed to be irreducible, then the proof becomes clear - we have $k\subset k(\alpha)\subset l$ with first extension being of degree $[L:K]$ and, since $[l:k]\leq [L:K],$ we have $l=k(\alpha),$ so $[l:k]= [L:K]$ which is the definition of unramified extension.
Any ideas how I could prove that $\overline{p}$ is irreducible? I have seen a variant of Hensel's lemma (which ironically does not seem to be equivalent to Hensel's lemma) that guarantees it, however I hoped that there could be an alternative approach without that.
Thank you.