We did a couple of things with monomial ideals:
1) if a monomial ideal is radical, then it's generated by squarefree monomials.
2) V(a monomial ideal) = a union of coordinate planes of various dimensions.
Case 2 of the weak Nullstellensatz: when the projection of V(I) to a coordinate line hits a finite set.
Algebraically, I intersect C[x_i] = < f(x_i) >, and the finite set is the set of roots.
We showed that I + < x_i - m > is still not the whole ring, if m is one of the roots.
So enlarge I, and repeat until done, at which point I = < x_i - m_i : i=1..n >, and that V(I) contains one point, (m_1,...,m_n).
We started the strong Nullstellensatz, but didn't finish it.
