Using Hensel lemma to find roots in valuation rings

Let $K$ be a p-adic field with absolute Galois group $\varGamma$. Let $O^{ur}_K$ be the ring of integers of $K^{ur}$. Then, for $n$ prime to $p$, the $\varGamma$-module $\mu_{n}$ is unramified. Hensel lemma(applied to finite unramified extensions of $K$) provides an exact sequence $$ 0\to \mu_n \to (O^{ur}_K)^* \stackrel{\cdot n}{\longrightarrow} (O^{ur}_K)^* \to 0 . $$

This is an example in a book I'm reading, my problem is that I can't see how the Hensel lemma is used to derive the surjectivity of $(O^{ur}_K)^* \stackrel{\cdot n}{\longrightarrow} (O^{ur}_K)^*$. It shold reduce to finding root of $x^n=a$ ($a$ invertible in $O^{ur}_K$) in the valuation ring $O^{ur}_K$, but I can't see how to carry out.

Thanks for any help.

Solution 1:

$P(X)=X^n-a$ is a monic polynomial which is separable mod the maximal ideal. The residue field of $K^{ur}$ is algebraically closed and thus $P$ has a root in the residue field. Taking a lift of this root, we find some $x \in O_{L}$ such that $P(x)$ is not invertible and $P’(x)$ is (invertible), where $L/K$ is a finite unramified extension. Then Hensel’s lemma shows that there is an $x’\in L$ equal to $x$ mod the maximal ideal such that $P(x’)=0$.