Exercise 2, § 3

Let $A$ be a noetherian integral ring. Suppose that $A$ is not a field and that the field of fractions of $A$ is a finitely generated $A$-algebra. Then $A$ is semi-local and of dimension 1. (Show that there exists a nonzero and noninvertible element $f$ of $A$ such that $A[f^{-1}]$ is a field, and reduce to proving that $\dim(A/(f)) = 0$; if this were not so, $A/(f)$ would possess a prime ideal of height 1 whose inverse image $\mathfrak{p}$ in $A$ would be a prime ideal of height 2; but the local ring $A_{\mathfrak{p}}$ of dimension 2 could have only a finite number of prime ideals, contrary to the prop. 6 of § 3, No. 3.)