Exercise 4, § 8

Let $k$ be a field, $K = k(X, Y)$ the field of rational functions in two indeterminates over $k$ and $v$ the valuation on $K$ with values in the group $\mathbf{Z} \times \mathbf{Z}$ ordered lexicographically, such that $v(X) = (0, 1)$, $v(Y) = (1, 0)$. Let $K'$ be the field $K(\sqrt{X})$; show that $v$ has a single extension $v'$ to $K$ and that $e(v'/v) = 2$, $f(v'/v) = 1$, but the ring of the valuation $v'$ is not a finitely generated module over the ring of the valuation v.