Definition of a radical ideal, and the radical of an ideal.
Proof of the easy half of the Nullstellensatz, that I(V(I)) contains the radical of I.
(The hard half is the opposite containment; we'll do that later.)
An algebraic subset is one of the form V(I). The Nullstellensatz says that they correspond 1:1 to radical ideals.
Proof that V(I+J) = V(I) intersect V(J).
Some examples of V(I intersect J), i.e., why it's interesting to write an ideal as an intersection.
