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Problem Set 3 Problem 1. a) Let I = (x4

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
k and all of A1
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


Source: Abo, Hirotachi - Department of Mathematics, University of Idaho


Collections: Mathematics