Nullstellensatz

Statement

Let be an algebraically closed field. Let

If vanishes on the common vanishing locus of the ‘s then some power of lies in .

Equivalent statement

If , then for the radical ,

Proof

This trick is often known as Rabinowitsch’s trick.

For we can also think of these polynomials as living in the space with one more variable, i.e. . Looking at this new polynomial ring with an extra variable, the set

has no common zeros. Thus, by weak Nullstellensatz says that which means . This gives

Consider the image of this expression under the map

So we have

Now we can multiply by some power of to “cancel denominators” to give