Basic example of unramified extension by take a root of a unit (not necessarily 1)

Question

Let $K$ be a local field with residue characteristic $p$, and let $u \in O_K$ be a unit. Let $e$ be relatively prime to $p$.

Let $L=K(u^{1/e})$ why is $L/K$ an unramified extension? Do we need $(e ,p)=1$?

Let $l, k$ be residue fields of $O_L, O_K$.

EDIT: Here is a simple argument I came up with, is it correct:
Let me assume (even though it might not be the case) $X^e-u$ is irreducible over $K$. Then in $k[X]$ since $X^e-\overline{u}$ is separable (since $(e, p)=1$), then it is irreducible (because if it factored, we could lift the factorization to $K$ by Hensel).
So $[L:K]$ and $[l:k]$ are equal since they both equal $e$, and that is a definition of unramified.

Answer 1

We do need $(e, p) = 1$. Also, your argument is correct.