Exercise A

• Prove that the two ideals I = <x > and J = <x - y²> in C[x,y] are radical.
• Prove that the sum I + J is not radical.

Exercise B

• Let F: kⁿ → k^m be a polynomial mapping (i.e. all coordinate functions of F are in k[x₁,...,x_n]). Let V be a subset of k^m. The inverse image is then defined as F^-1 = {a ∈ kⁿ | F(a) ∈ V}. Prove that F^-1(V) is a variety if V is a variety.
• Now let W ⊆ kⁿ be a variety. The image of F is the set F(V) = { F(a) | a ∈ V}. Prove that if F(V) is a variety then it is irreducible if V is irreducible.