In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients). A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain , "rational numbers" must be replaced by "field of fractions of ". This implies that, if is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials. This is systematically used (explicitly or implicitly) in all implemented algorithms (see Polynomial greatest common divisor and Factorization of polynomials). Gauss's lemma, and all its consequences that do not involve the existence of a complete factorization remain true over any GCD domain (an integral domain over which greatest common divisors exist). In particular, a polynomial ring over a GCD domain is also a GCD domain. If one calls primitive a polynomial such that the coefficients generate the unit ideal, Gauss's lemma is true over every commutative ring. However, some care must be taken when using this definition of primitive, as, over a unique factorization domain that is not a principal ideal domain, there are polynomials that are primitive in the above sense and not primitive in this new sense. The lemma over the integers If is a polynomial with integer coefficients, then is called primitive if the greatest common divisor of all the coefficients is 1; in other words, no prime number divides all the coefficients. Proof: Clearly the product f(x)g(x) of two primitive polynomials has integer coefficients. Therefore, if it