# Online Encyclopedia

# Gauss lemma

In mathematics, there is more than one **Gauss lemma**; all are named after Carl Friedrich Gauss.

In the theory of polynomials, the **Gauss lemma** relates the highest common factor of a product of two polynomials with integer coefficients to the corresponding hcfs of the factors. If we are looking at *R* = *P*.*Q* then any common factor of the coefficients of *P* will divide all the coefficients of *R*, by an easy proof. The lemma works the other way round, limiting the common factors for *R*. It supplies what is needed to conclude that the hcf of the coefficients of *R* is exactly the product of the hcfs for *P* and for *Q*.

A statement that is equivalent: if the hcf for *P* and for *Q* is 1, then it is 1 for *R*, also.

In the case of one variable there is a simple proof of this. Consider a prime number *p*, and try to show that *R* mod *p* (i.e. R with coefficients reduced to the field of residues modulo *p*) is not 0. In fact the degree of *R* mod *p* is the sum of those of *P* mod *p* and of *Q* mod *p*, which is more than enough, because we are working in a field.

An important consequence is that *R* can only factorise as a product of polynomials with rational number coefficients, if it already does into integer polynomials. One sees this by checking the powers of a fixed prime *p* needed to clear denominators; the same argument works as before, and this version can also be called the Gauss lemma. It applies to the rational root theorem.

There is a generalisation to several variables.

The **Gauss lemma** in number theory is involved in some proofs of quadratic reciprocity.

For any odd prime *p* let *a* be an integer that is relatively prime to *p*.

Consider the integers

and their least positive residues modulo *m*.

Let *n* be the number of these residues that are greater than *p/2*. Then

where *(a/p)* is the Legendre symbol.

This can, for example, be applied immediately when *a* = −1, giving

*n*= ½(*p*− 1).

From a sophisticated point of view, this is a case of the transfer.