Finite extension of local field.

Question

Let $F$ be a finite extension of $\Bbb{Q}_p$. Let K be tamely ramified extension of $F$ Containing the maximal unramified extension. Let $P$ denote the residue field of the corresponding tamely ramified extension. Is there an easy way of showing that the natural map $(O_K)\rightarrow P$, is injective on the roots of unity lying in $K$.

Answer 1

The extension $\Bbb Q_p(\zeta_p)\supset\Bbb Q_p$ is tamely ramified (degree is $p-1$), and the $p$-th roots of unity are sent to the identity in $\Bbb F_p$. (Perhaps I misunderstand your question?)