# Rabinowitsch trick

In mathematics, the **Rabinowitsch trick**, introduced by George Yuri Rainich and published under his original name Rabinowitsch (1929),
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 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 lies in the ideal generated by *f*_{1},....,*f*_{m}. This is the full version of the Nullstellensatz for *K*[*x*_{1},...,*x*_{n}].

## References

- Brownawell, W. Dale (2001) [1994], "Rabinowitsch trick", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Rabinowitsch, J.L. (1929), "Zum Hilbertschen Nullstellensatz",
*Math. Ann.*(in German),**102**(1): 520, doi:10.1007/BF01782361, MR 1512592