Exercise 5, § 1

Let $\mathfrak{P}$ be the set of prime numbers and put $A = \prod_{p \in \mathfrak{P}} F_p$. For every filter $\mathscr{F}$ on $\mathfrak{P}$ (Gm. Top., I, p. 57) denote by $m_{\mathscr{F}} \subset A$ the set of $u = (u_p)_{p \in \mathfrak{P}}$ such that the set of $p \in \mathfrak{P}$ with $u_p = 0$ is a member of $\mathscr{F}$. Show that $m_{\mathscr{F}}$ is an ideal and that we thus obtain a bijection between the ideals of $A$ distinct from $A$ and the filters on $\mathfrak{P}$. Let $\mathscr{F}$ be an ultrafilter (Gen. Top., I, p. 60) on $\mathfrak{P}$ which does not possess a limit in the discrete topology. Show that $A / m_{\mathscr{F}}$ is a field of characteristic 0, isomorphic to a subfield of $\mathbf{C}$. \*