Polynomials giving integer outputs

Question

Given a polynomial $p(x)$ that outputs an integer for every integer input $x$, is it necessary for $p(x)$ to have rational coefficients?

Answer 1

No, say $$p(x) = {1\over 2}x^2+{1\over 2}x$$

or for every natural number $n$: $$p(x) ={x+n\choose n}$$