 
Summary: Problem Set 3
Problem 1. a) Let I = (x4
 1) R[x]. Find I(V (I)).
b) Let I = (x5
 1) R[x]. Find I(V (I)).
c) Let I = (xn
 1) R[x]. Find I(V (I)).
Problem 2. Give an example of 2 ideals I and J in C[x, y] such that I = J but
V (I) = V (J).
Problem 3. Show that the closed ane varieties in A1
k are the nite subsets of
A1
k and all of A1
k.
I thank George Raptis for posing the following question:
Problem 4. Show that if k is a nite eld, then EVERY subset of An
k is a closed
ane variety. (Hence we can conclude that the Zariski topology on An
k is the
same as the discrete topology on An
