Corollary 1

Let K be a commutative field, A be a commutative K-algebra generated by finitely many elements, and $\mathfrak{m}$ be a maximal ideal of A.

a) The degree of $A/\mathfrak{m}$ over K is finite.

b) Let Ω be an algebraically closed extension of K. There exists a K-algebra homomorphism from A to Ω with kernel $\mathfrak{m}$.

The K-algebra $A/\mathfrak{m}$ is generated by finitely many elements, and $A/\mathfrak{m}$ is a field. By Theorem 1, the degree of $A/\mathfrak{m}$ is finite; assertion a) follows. Every extension of K of finite degree is isomorphic to a subextension of Ω (V, §4, No. 1, p. 20, Theorem 1); this gives b).