I am trying to prove the local Kronecker-Weber theorem for tamely ramified abelian extensions $L|\mathbb{Q}_p$. At some point in the proof I need to show that $\mathbb{Q}_p(u^{1/e})$ is unramified under the assumptions that $u$ is a unit (possibly in some extension field) and $(e,p)=1$. Supposedly the following argument works: Since $e$ and $p$ are coprime and $u$ is a unit, the discriminant of $x^e-u$ is not divisible by $p$ and hence the splitting field is unramified.
I can't see why the discriminant is not divisible by $p$. I have expanded the discriminant but keep running into an issue where the number of terms may be divisible by $p$,.
One way to think about is that you want to check if the disc. is divisible by $p$, so it suffices to compute the disc. modulo $p$. Over a field, the disc. is a unit if and only if $f$ and $f'$ are coprime, so you have to determine whether $x^e - u$ and $e x^{e-1}$ (thought of modulo $p$) are coprime. Since $u$ is a unit by assumption, this comes down to whether or not $e$ is a unit mod $p$, i.e. whether or not $p$ divides $e$.