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 νp(x). For a polynomial Q with integer coefficients, the sequence of valuations νp(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.