n-th power residues in p-adic fields How to see that number of n-th power residue classes of a p-adic field K (elements of $K^*/K^{*n}$) is finite?
I know how to prove using Hensel lemma that all p-adic integers sufficiently close to 1 are n-powers but cannot apply it to prove the statement.

Another approach, described a bit sketchily.
Let $\pi$ be a prime element, i.e. generator of the maximal ideal of the ring $\mathfrak o_K$ of integers of $K$. Then $K^*=\pi^{\Bbb Z}\oplus\mathfrak o_K^*$, where $\mathfrak o_K^*$ is the group of units, which is compact. But $z\mapsto z^n$ is an open mapping, you’ve already proved this, and so the image is open in the compact group $\mathfrak o_K^*$, consequently of finite index.
