I was reading about Gauss's Lemma here: https://cims.nyu.edu/~kiryl/Algebra/Section_3.10--Polynomials_Over_The_Rational_Field.pdf Unfortunately, I am stuck on Lemma 3.10.1 that concludes that the product of a pair of primitive polynomials is itself primitive. I understand about how there is some index in each polynomial at which the polynomial becomes imprimitive, and that the term cj + k is of the form: cj + k = aj bk + Ta ga + Tb gb where ga & gb are the Greatest Common Divisor of the terms below aj & bk, respectively (here, I use Ta & Tb to denote general numbers that are multiplied by ga & gb to get the terms for the summation of the products of a & b terms whose index is less than i & j, respectively) So the proof follows that it presumes that cj + k has a prime divisor. Now, if I presume that this divisor is the same as ga & gb, then I get the contradiction that cj + k has no divisor. Fine. However if I only presume that the divisor is the same as ga, but not gb, I get (here U is a general number like T) cj + k = ga U = aj bk + Ta ga + Tb gb aj bk = ga ( U - Ta ) - gb Tb Now I don't know what prime factors aj or bk have, and it seems that I have no way of proving that the RHS of this doesn't coincidentally sum up to a number that happens to be a multiple of whatever prime factors aj or bk What am I missing here?