I'm stuck with an article "A simple proof of Kronecker-Weber Theorem" on this website. On page 7, the author proofs that $\mathbb{Q}_p((-p)^{\frac{1}{p-1}}) = \mathbb{Q}_p(\zeta_p)$. While I understand the reasoning of the proof ($[\mathbb{Q}_p((-p)^{\frac{1}{p-1}}):\mathbb{Q}_p] = [\mathbb{Q}_p(\zeta_p):\mathbb{Q}_p]$, so if one contains the other, we are done), I don't get how he makes his $u$ to use for Hensel's Lemma.
Specifically, he defines a polynomial $g(X) = \frac{(X+1)^p-1}{X} = X^{p-1}+pX^{p-2}+\ldots +p,$ so of course we have $g(\zeta_p-1)=0$, but I don't get how $g(\zeta^p-1)\equiv (\zeta_p-1)^{p-1}+p (\bmod(\zeta_p-1)^p).$ Apart from understanding this lemma, is there an other (simpler) proof of this fact ?
Any help would be appreciated.