Hilbert's Nullstellensatz

The Hilbert Nullstellensatz is in mathematics in classical algebraic geometry, the central connection between ideals and affine algebraic varieties ago. It was proved by David Hilbert. There are several equivalent variants to formulate the Nullstellensatz:

• Is an algebraically closed field and a proper ideal, so there is a so that

• Is an algebraically closed field and an ideal in, then:

• Of the radical,
• The set of all common zeros of (as above), and
• The ideal of all polynomials that vanish on.

• It was a body and a maximal ideal in. Then, the degree of the field extension is finite.

• It should be an algebraically closed field and a maximal ideal in. Then, for a point

• It was a body and a body extension that is as - algebra finitely generated. Then is finite; in particular, the extension is algebraic.

It follows from the Hilbert- between Nullstellensatz that the pictures and for a algebraically closed field is a bijective relationship between affine algebraic sets and radical ideals in define. This can be restricted to bijective relations between irreducible algebraic quantities and prime ideals and between points in and maximal ideals.