The Stacks project

Remark 111.16.2. Let $k$ be a field. Then for every integer $n\in {\mathbf N}$ and every maximal ideal ${\mathfrak m} \subset k[x_1, \ldots , x_ n]$ the quotient $k[x_1, \ldots , x_ n]/{\mathfrak m}$ is a finite field extension of $k$. This will be shown later in the course. Of course (please check this) it implies a similar statement for maximal ideals of finitely generated $k$-algebras. The exercise above proves it in the case $k = {\mathbf C}$.