 Solution

ID: past-exam-of-the-algebraic-geometry-course-of-the-university-of-oxford/2011/1/b/solution

I = {f \in R : f^{N} \in I for some integer N \geq 1} .

I = I

. Its zero set is

V (I) = {a \in k^{n} : f (a) = 0 for every f \in I}

Every

f \in I

vanishes at every point of

V (I)

, so

I \subseteq I (V (I))

. If

g^{N} \in I (V (I))

, then

g (a)^{N} = 0

in the field

k

for every

a \in V (I)

, so

g (a) = 0

for each such point. Hence

g \in I (V (I))

and

I (V (I)) is a radical ideal containing I .

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