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We determine the probability that a random polynomial of degree $n$ over$\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it isgiven by a rational function of $p$ that is invariant under replacing $p$ by$1/p$.

The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$