Exercise 4, § 3

Let A be a local integral domain, K its field of fractions, m its maximal ideal and $v$ a valuation on K whose ring B dominates A. Suppose that, either m is a finitely generated ideal, or $v$ is a discrete valuation; then there exists in m an element $x$ such that $v(x) = \inf_{y \in m} v(y)$. Let A' be the subring of K generated by A and the elements $y x^{-1}$, where y runs through m; let A, be the local ring of A' relative to the prime ideal $p \cap A'$, where p is the ideal of the valuation $u$. Show that the ring A, does not depend on the choice of the element $x$ satisfying the above conditions (A, is called "the prime quadratic transform of A in the direction of $v$"); if A is Noetherian, so is A,.