Exercise 11, § 2

Let $P$ be the set of prime numbers, $P'$ an infinite subset of $P$, $f \in \mathbf{Z}[(X_i)]$. For every $p \in P'$, 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 $\mathbf{Z}/p\mathbf{Z}$. We suppose that, for every $p \in P'$ and every $x \in (\mathbf{Z}/p\mathbf{Z})^I$, we have $f_p(x) = 0$. Then $f = 0$.