Exercise 11, § 2

Let $P$ be the set of all prime numbers, $P'$ an infinite subset of $P$ and $f \in Z[(X_i)]$. For each $p \in P'$ let $f_p$ be the element of $(\mathbf{Z}/p\mathbf{Z})[(X_i)]$ obtained by applying to the coefficients of $f$ the canonical homomorphism of $\mathbf{Z}$ onto $\mathbf{Z}/p\mathbf{Z}$. Suppose that for each $p \in P'$ and each $x \in (\mathbf{Z}/p\mathbf{Z})'$ we have $f_p(x) = 0$; then $f = 0$.