Exercise 1, § 7

Let $(\phi_i)_{i \in I}$ be a family of places of a field $K$ taking their values in a finite number of fields; suppose further that for some $x \in K$, the set of $\phi_i(x)$ is finite. Show that there exists a polynomial $f(X)$ of the form (1) of no. 1 such that $f(x) \neq 0$ and that the element $z = f(x)^{-1}$ enjoys the following properties: (i) $\phi_i(x) = \infty$ implies $\phi_i(xz) = 0$ and $\phi_i(z) = 0$; (ii) $\phi_i(x) \neq \infty$ implies $\phi_i(xz) \neq \infty$ and $\phi_i(z) \neq 0$. (Same proof as for Lemma 1).