Solutions in $\mathbb Q_p$ leads to solution for congruences equations?

Question

Let $p$ be a prime number such that $p\equiv1\pmod 3$. Let $n$ be an integer such that the equation $x^3=n$ has a solution in $\mathbb Q_p$. In fact with our assumptions, the others solution are in $\mathbb Q_p$ too. So the polynomial $X^3-n$ splits in linear factors in $\mathbb Q_p$. Can one deduce that the equation $X^3\equiv n\pmod {p^\alpha}$ is solvable (in $\mathbb Z$) for every $a\in\mathbb N^*$?