Math 8203
Fall 1997

Hint for Exercise 8(d)

Here are suggestions for two of the main steps:

• Suppose that we have an equality V(f) = YZ, where Y and Z are nonempty varieties. Show that this leads to an inclusion: V(f) V(g) V(h), where g and h are nonzero nonconstant polynomials. Show that if Y and Z are proper subsets of V(f), then we can find g and h with the additional property that that neither V(g) nor V(h) contains V(f).
• If f, g, and h are as above, use the (geometric) information from previous parts of this exercise to deduce the (algebraic) conclusion that either g or h must be divisible by f
  [and explain why this leads to a contradiction].

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