p-adic cubic root

Question

Let $p$ be prime such that $p\equiv 2\bmod 3$. Show that for every $a\in \mathbb Z,p\nmid a$ there is a $x\in \mathbb Z_p$, where $\mathbb Z_p$ is the field of the p-adic integers, such that $x^3=a$.

Answer 1

Hint (already given in comments): Hensel's lemma reduces to showing all elements of $\Bbb F_p^\times$ are cubes, which follows easily from $3\nmid(p-1)$. Can you see why?

Perhaps you'd get what's going on if I state it in a more general form: if $G$ is a finite group with order $n$ and $m$ is any number coprime to $n$, then $x\mapsto x^m$ must be a bijection on $G$. (Why?)