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mathwonk
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E.g. one version of the nullstellensatz is that every maximal ideal in C[X1,...,Xn] is the ideal of a single point, say p = (a1,...,an), hence has form (X1-a1,...Xn-an). You might think this is obvious, since a maximal ideal must correspond to a minimal variety, and surely one point is a minimal variety. But no, the empty set is even smaller! so the whole point of the nullstellensatz is to prove that the variety of every proper ideal is non empty.