In mathematics, the Rabinowitsch trick, introduced by , 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 such that as an equality of elements of the polynomial ring. Since 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 f^r lies in the ideal generated by f1,....,fm. This is the full version of the Nullstellensatz for K[x1,...,xn].