Exercise 6, § 2

Let $A$ be an integrally closed ring, $K$ its field of fractions, $\mathfrak{p}$ a prime ideal of $A$, $k$ the field of fractions of $A/\mathfrak{p}$. Prove now that if $A_{\mathfrak{p}}$ is not a valuation ring, there exists a valuation ring $V$ for $K$ which dominates $A_{\mathfrak{p}}$ and whose residue field is a transcendental extension of $k$ (use Chapter V, § 2, Exercise 14 b)).