We have,
$\mathbb{Q}_p=$p-adic field, $\mathbb{Z}_p=$ring of p-adic integers, $\mathbb{Z}_p^{\times}=$multiplicative group of units in $\mathbb{Z}_p$.
We have the following decompositions:
$\mathbb{Z}_p^{\times}=\mu_{p-1} \times (1+p\mathbb{Z}_p)$,
where $\mu_{p-1}$ is the group of roots of unity of order $ \ p-1$,
$\mathbb{Q}_p^{\times}=p^{\mathbb{Z}} \times \mu_{p-1} \times (1+p\mathbb{Z}_p) \simeq \mathbb{Z} \times \mathbb{Z} /(p-1) \mathbb{Z} \times \mathbb{Z}_p$.
The question is-
How can we have he similar decomposition of $ \ \mathbb{Q}_p[\zeta_p]^{\times}$ and $\mathbb{Q}_p(\zeta_p)$ ?