True. Consider the polynomial factorised into its roots: (x-a)(x-b)...(x-z) where a, b, z, etc. are all in Z, then the expansion of the product of the factors will contain the products, sums of products, of combinations of a, b, ...z. But these quantities are, by definition, contained in Z. The products and sums of products form the coefficients in the expansion, therefore the coefficients are all in Z.