Implications
The first result implies that the contents of polynomials, defined as the GCD of their coefficients, are multiplicative: the contents of the product of two polynomials is the product of their individual contents.
The second result implies that if a polynomial with integer coefficients can be factored over the rational numbers, then there exists a factorization over the integers. This property is also useful when combined with properties such as Eisenstein's criterion.
Both results are essential in proving that if R is a unique factorization domain, then so is R (and by an immediate induction, so is the polynomial ring over R in any number of indeterminates). For any factorization of a polynomial P in R, the statements imply that the product Q of all irreducible factors that are not contained in R (the non-constant factors) is always primitive, so P = c(P)Q where c(P) is the contents of P. This reduces proving uniqueness of factorizations to proving it individually for c(P) (which is given) and for Q. By the second statement the irreducible factors in any factorization of Q in R are primitive representatives of irreducible factors in a factorization of Q in F, but the latter is unique since F is a principal ideal domain and therefore a unique factorization domain.
The second result also implies that the minimal polynomial over the rational numbers of an algebraic integer has integer coefficients.
Read more about this topic: Gauss's Lemma (polynomial)
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