  We study the roots of a random polynomial over the field of p-adic numbers.
For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an
asymptotic formula for the factorial moments of the number of roots of this
polynomial. In addition, we show the probability that a random polynomial of
degree $n$ has more than $\log n$ roots is $O\big(n^{-K}\big)$ for some $K >
0$.
