Exercise 24, § 1

Let $K$ be an algebraically closed field and $A = K[X, Y]$ the polynomial ring in two indeterminates over $K$, which is a Krull domain \* (and even a factorial domain). \* For every ordered pair $(\alpha, \beta) \in K^2$ let $w_{\alpha, \beta}$ be the discrete valuation on $A$ such that, for every polynomial $f \neq 0, w_{\alpha, \beta}(f)$ is the least degree of the monomials $\neq 0 \inf (X + \alpha, Y + \beta)$. Show that $A$ is the intersection of the rings of the valuations $w_{\alpha, \beta}$, none of which is an essential valuation of $A$.