Hilbert’s Nullstellensatz

Abstract

Hilbert’s Nullstellensatz may be seen as the starting point of algebraic geometry. It provides a bijective correspondence between affine varieties, which are geometric objects, and radical ideals in a polynomial ring, which are algebraic objects. In this chapter, we give proofs of two versions of the Nullstellensatz. We exhibit some further correspondences between geometric and algebraic objects. Most notably, the coordinate ring is an affine algebra assigned to an affine variety, and points of the variety correspond to maximal ideals in the coordinate ring.