Hilbert's Nullstellensatz

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Hilbert's Nullstellensatz (German: "theorem of zeros") is a theorem in algebraic geometry, a branch of mathematics, that relates varieties and ideals in polynomial rings over algebraically closed fields. It was first proved by David Hilbert.

Let K be an algebraically closed field (such as the complex numbers), consider the polynomial ring K[X₁,X₂,..., X_n] and let I be an ideal in this ring. The affine variety V(I) defined by this ideal consists of all n-tuples x = (x₁,...,x_n) in Kⁿ such that f(x) = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in K[X₁,X₂,..., X_n] which vanishes on the variety V(I), i.e. p(x) = 0 for all x in V(I), then there exists a natural number r such that p^r is in I.

An immediate corollary is the "weak Nullstellensatz": if I is a proper ideal in K[X₁,X₂,..., X_n], then V(I) cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal. This is the reason for the name of the theorem, which can be proved easily from the 'weak' form. Note that the assumption that K be algebraically closed is essential here: the proper ideal (X² + 1) in R[X] does not have a common zero.

With the notation common in algebraic geometry, the Nullstellensatz can also be formulated as

for every ideal J. Here, denotes the radical of J and I(U) is the ideal of all polynomials which vanish on the set U. In this way, we obtain an order-reversing bijective correspondence between the affine varieties in Kⁿ and the radical ideals of K[X₁,X₂,..., X_n].