Exercise 5, § 3

Let A be a commutative algebra and let S be a subset of A generating A as K-algebra. Suppose that $\operatorname{Card}(S) < \operatorname{Card}(K)$. Show that if $\mathfrak{M}$ is a maximal ideal of A, then the field $A/\mathfrak{M}$ is an algebraic extension of K. (Use Ex. 15 of V, p. 174 when $\operatorname{Card}(K) \leq \operatorname{Card}(N)$ and Ex. 4 when

Deduce that $A/\mathfrak{M} = K$ when K is algebraically closed.