  The most frequently asked question in the $p-$adic lattice models of
statistical mechanics is that whether a root of a polynomial equation belongs
to domains $\mathbb{Z}_p^{*}, \ \mathbb{Z}_p\setminus\mathbb{Z}_p^{*}, \
\mathbb{Z}_p, \ \mathbb{Q}_p\setminus\mathbb{Z}_p^{*}, \
\mathbb{Q}_p\setminus\left(\mathbb{Z}_p\setminus\mathbb{Z}_p^{*}\right), \
\mathbb{Q}_p\setminus\mathbb{Z}_p, \ \mathbb{Q}_p $ or not. However, this
question was open even for lower degree polynomial equations. In this paper, we
give local descriptions of roots of cubic equations over the $p-$adic fields
for $p>3$.
