Past exam of the introduction to schemes course of the University of Oxford / 2023 / 2 / c / Solution

Let $x\in \mathrm{Frac}(D)$ satisfy a monic polynomial equationIf $x=0$, there is nothing to prove. Suppose $x\notin D$. The defining property of a valuation ring gives $y=x^{-1}\in D$. Multiplying the equation by $y^{n-1}$ yieldsa contradiction. ThereforeThis proof uses only the comparison of an element with its inverse; no discreteness or finite generation of the valuation is needed.