Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite.

Question

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite.

$P$ is a maximal ideal of $R/P$ is a field. I know that the result directly follows from the weak nullstellensatz if we can show that $R/P$ is a finitely generated $F$-algebra, but how can we be sure that $R/P$ contains $F$?

Answer 1

We have a homomorphism $F \to R \to R/P$. A homomorphism between fields is automatically a field extension, hence $R/P$ contains $F$.