Assuming $\zeta_p$ is a root of unit, I need to show that $\cal{O}$$(\mathbb{Q}_p(\zeta_p))=\mathbb{Z}_p [\zeta_p]$. Where $\cal{O}$$(F)=${$x\in F:|x|_p\le 1$} and $\mathbb{Z}_p$ is the ring of p-adic intageres.
Note that: $\mathbb{Q}_p(\zeta_p)$ is a totally ramified extention of $\mathbb{Q}_p$ of degree $p-1$
I did manage to show that $\cal{O}$$(\mathbb{Q}_p(\zeta_p))\supset\mathbb{Z}_p [\zeta_p]$, but not the other way.
I'd love to get some help