Exercise 6, § 3

Let C be an uncountable algebraically closed field and $P = C[(X_i)_{i \in I}]$ a polynomial algebra over C. Given two families $(f_\alpha)_{\alpha \in A}, (g_\beta)_{\beta \in B}$ of elements of P suppose that: a) B and I are countable. b) For every finite subset $A'$ of A and every finite subset $B'$ of B there exists $(x_i)_{i \in I} \in C^I$ such that

Show that then there exists $(x_i)_{i \in I} \in C^I$ such that

(Let $(Y_\beta)_{\beta \in B}$ be a family of indeterminates indexed by B and let Q be the polynomial algebra $\mathbf{C}[(X_i)_{i \in I}, (Y_\beta)_{\beta \in B}]$. Let A be the quotient of Q by the ideal generated by the $f_\alpha$ and the $1 - g_\alpha Y_\alpha$. Show that $A \neq 0$ thanks to a) and b) and apply Ex. 5 to a maximal ideal of A.) \*