 pth-power criterion for p-adic units

For an odd prime $p$ and $\alpha \in {\mathbb{Z}}_p^{\times }$,This follows from ${\mathbb{Z}}_p^{\times }\cong {\mu }_{p-1}\times (1+p{\mathbb{Z}}_p)$ and $(1+p{\mathbb{Z}}_p{)}^p=1+p^2{\mathbb{Z}}_p$.