 Solution

ID: past-exam-of-the-introduction-to-schemes-course-of-the-university-of-oxford/2023/2/c/solution

Let

x \in Frac (D)

x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{0} = 0, a_{i} \in D .

x = 0

, there is nothing to prove. Suppose

x \in / D

. The defining property of a valuation ring gives

y = x^{- 1} \in D

. Multiplying the equation by

y^{n - 1}

yields

x = - a_{n - 1} - a_{n - 2} y - \dots - a_{0} y^{n - 1} \in D,

D is an integrally closed domain .

This proof uses only the comparison of an element with its inverse; no discreteness or finite generation of the valuation is needed.

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