Periodicity in the $p$-adic valuation of a polynomial

For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by ${\nu }_p(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations ${\nu }_p(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic situation, the period length is determined.