**Exercise A**
- Prove that the two ideals
*I = <x >* and *J
= <x - y*^{2}> in *C[x,y]* are radical.
- Prove that the sum
*I + J* is not radical.

**Exercise B**
- Let
*F: k*^{n} → k^{m} be a
polynomial mapping (i.e. all coordinate functions of *F*
are in *k[x*_{1},...,x_{n}]). Let
*V* be a subset of *k*^{m}. The
**inverse image** is then defined as
*F*^{-1} = {*a ∈ k*^{n} |
*F(a) ∈ V*}. Prove that *F*^{-1}(V)
is a variety if *V* is a variety.
- Now let
*W ⊆ k*^{n} 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.