Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name ,
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"["x"1..."x""n"] vanishes whenever all polynomials "f"1...,"f""m" vanish. Then the polynomials "f"1...,"f""m", 1 − "x"0"f" have no common zeros (where we have introduced a new variable "x"0), so by the weak Nullstellensatz for "K"["x"0, ..., "x""n"] they generate the unit ideal of "K"["x"0 ..., "x""n"]. Spelt out, this means there are polynomials formula_1 such that
as an equality of elements of the polynomial ring formula_3. Since formula_4 are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting formula_5 that
as elements of the field of rational functions formula_7, the field of fractions of the polynomial ring formula_8. 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 formula_10. Hence
which literally states that formula_12 lies in the ideal generated by "f"1...,"f""m". This is the full version of the Nullstellensatz for "K"["x"1...,"x""n"].