I'm following the proof given in Koblitz's book which roughly speaking builds the un-ramified extension of degree $f$ of $\mathbb{Q}_p$ as $\mathbb{Q}_p(\alpha)$, where $\alpha$ is a root of the lift of the minimal polynomial of a primitive root in $\mathbb{F}_{p^f}$.
Can we assume that $\alpha$ is an integral unit?
I think yes, and my idea is to use symmetricity of the minimal polynomial of a primitive root in $\mathbb{F}_{p^f}$, which I think can be preserved after the lifting, (hence $\alpha^{-1}$ is an integer and therefore is a unit) but being these my firsts steps in this field (pun intended) I need some guidance and opinion.