1. Let I be a radical ideal, and {g_1..g_m} a reduced Gr\"obner basis for it.
Show that each g_i is squarefree, i.e. is not divisible by f^2 for any polynomial f of degree > 0.
2. Let I be an ideal such that for all f, if f^2 is in I, then f is in I. Show that I is radical.
3. Let I be a monomial ideal generated by squarefree monomials. Show that I is radical.
Hint: show that if p is in I, then init(p) and p-init(p) are in I. Use this to show that it is enough to test the condition in question #2 when f is a monomial.
4. Let I be an ideal with a Gr\"obner basis {f_1,...,f_n}, such that each init(f_i) is a squarefree monomial. Show that I is radical.
5. Let I and J be two radical ideals.
a) Show that I intersect J is radical.
b) Give an example where I+J (which concatenates their generators) is not radical.
6. Let I = < ac,bc,bd,ae,de > inside C[a,b,c,d,e]. Decompose V(I) as a union of subspaces. Make it minimal, so no subspace in your list contains another.
