is not primitive, there must be a prime p which is a common divisor of all its coefficients. But p can not divide all the coefficients of either f(x) or g(x) (otherwise they would not be primitive). Let arxr be the first term of f(x) not divisible by p and let bsxs be the first term of g(x) not divisible by p. Now consider the term xr+s in the product, whose coefficient is The term arbs is not divisible by p (because p is prime), yet all the remaining ones are, so the entire sum cannot be divisible by p. By assumption all coefficients in the product are divisible by p, leading to a contradiction. Therefore, the coefficients of the product can have no common divisor and are thus primitive. The proof is given below for the more general case. Note that an irreducible element of Z (a prime number) is still irreducible when viewed as constant polynomial in Z[X]; this explains the need for "non-constant" in the statement. Statements for unique factorization domains Gauss's lemma holds more generally over arbitrary unique factorization domains. There the content of a polynomial can be defined as the greatest common divisor of the coefficients of (like the gcd, the content is actually a set of associate elements). A polynomial with coefficients in a UFD is then said to be primitive if the only elements of that divide all coefficients of at once are the invertible elements of ; i.e., the gcd of the coefficients is one. Primitivity statement: If is a UFD, then the set of primitive polynomials in is closed under multiplication. More generally, the content of a product of polynomials is the product of their individual contents. Irreducibility statement: Let be a unique factorization domain and its field of fractions. A non-constant polynomial in is irreducible in if and only if it is both irreducible in and primitive in . (For the proofs, see #General version below.) Let be a unique factorization domain with field of fractions . If is a polynomial over then for some in , has coefficients in , and so – factoring out the gcd of the coefficients – we can write for some primitive polynomial . As one can check, this polynomial is unique up to the multiplication by a unit and is called the primitive part (or primitive representative) of and is denoted by .