Math 176 algebraic geometry fall 2014- assignment 2 for

Question

Math 176: Algebraic Geometry, Fall 2014- Assignment 2

1 For each of the following, determine if the statement is true or false. If true, give a proof. If false, give a counterexample.

(a) C[x, y] is a PID.

(b) If R is a UFD, and a, b ∈ R such that (a) = (b), then a = ub for some unit u ∈ R.

(c) If R is a PID and I is a prime ideal, then R/I is a PID.

2. (a) Let I = (x² + 1) in the ring C[x]. Prove C[x]/I ≅ C × C.

(b) Let I = (x² + 1, x - 2) in Z[x]. Prove Z[x]/I ≅ Z/5Z.

3. Let X = {(t, t², t³) ∈ A³(C): t ∈ C}. Show that I(X) = J where J = (z - x³, y - x²).

4. For each of the following, determine if the statement is true or false. If true, give a proof. If false, give a counterexample. (You may assume the field k has characteristic zero).

(a) If X, Y ⊆ Aⁿ, then X = Y if and only if I(X) = I(Y).

(b) If X,Y ⊆ Aⁿ are algebraic sets, then X = Y if and only if I(X) = I(Y).

(c) If J is an ideal in k[x₁, . . . , x_n], then rad(J) ⊆ I(V(J)).

(d) If J is an ideal in k[x1, . . . , xn], then rad(J) = I(V(J)).