for the GCD case). If , then there is nothing to prove. So, assume otherwise; then there is a non-unit element dividing the coefficients of . Factorizing that element into a product of prime elements, we can take that element to be a prime element . Now, we have: . Thus, either contains or ; contradicting the gcd's of the coefficients of are both 1. Remark: Over a GCD domain (e.g., a unique factorization domain), the gcd of all the coefficients of a polynomial , unique up to unit elements, is also called the content of . Applications It follows from Gauss's lemma that for each unique factorization domain , the polynomial ring is also a unique factorization domain (see #Statements for unique factorization domains). Gauss's lemma can also be used to show Eisenstein's irreducibility criterion. Finally, it can be used to show that cyclotomic polynomials (unitary units with integer coefficients) are irreducible. Gauss's lemma implies the following statement: If is a monic polynomial in one variable with coefficients in a unique factorization domain (or more generally a GCD domain), then a root of that is in the field of fractions of is in . If , then it says a rational root of a monic polynomial over integers is an integer (cf. the rational root theorem). To see the statement, let be a root of in and assume are relatively prime. In we can write with for some . Then is a factorization in . But is primitive (in the UFD sense) and thus divides the coefficients of by Gauss's lemma, and so with in . Since is monic, this is possible only when is a unit. A similar argument shows: Let be a GCD domain with the field of fractions and . If for some polynomial that is primitive in the UFD sense and , then . The irreducibility statement also implies that the minimal polynomial over the rational numbers of an algebraic integer has integer coefficients. Notes References Theorems about polynomials Theorems in ring theory Lemmas in algebra