Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by J.L.

Rabinowitsch (1929),[1] is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called weak Nullstellensatz), by introducing an extra variable.

The Rabinowitsch trick goes as follows.

Let K be an algebraically closed field.

Suppose the polynomial f in K[x1,...xn] vanishes whenever all polynomials f1,....,fm vanish.

Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the weak Nullstellensatz for K[x0, ..., xn] they generate the unit ideal of K[x0 ,..., xn].

Spelt out, this means there are polynomials

g

0

g

1

, … ,

such that as an equality of elements of the polynomial ring

are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting

that as elements of the field of rational functions

, the field of fractions of the polynomial ring

Moreover, the only expressions that occur in the denominators of the right hand side are f and powers of f, so rewriting that right hand side to have a common denominator results in an equality on the form for some natural number r and polynomials

Hence which literally states that

lies in the ideal generated by f1,....,fm.

This is the full version of the Nullstellensatz for K[x1,...,xn].