Hilbert's Nullstellensatz (German: "theorem of zeros") is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. It was first proved by David Hilbert. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...

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. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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. In mathematics, the empty set is the set with no elements. ...

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

I(V(J)) = √J    for every ideal J

Here √J denotes the radical of J and I(U) denotes 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]. In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...