Solvability of cubic equations in $p$-adic integers ($p>3$)

We give a criterion for the existence of solutions to an equation of the form $^3+ax=b$, where $a,b\in\mathbb Q_p$, in $p$-adic integers for $p>3$. Moreover, in the case when the equation $x^3+ax=b$ is solvable, we give necessary and sufficient recurrent conditions on a $p$-adic number $x\in\mathbb Z^*_p$ under which $x$ is a solution to the equation.