Gauss's lemma

Gauss's lemma is the bridge that lets you carry the good factorization behavior of a ring up to its polynomials, and lets you study factoring polynomials over a domain by working over the easier field of fractions. It is the technical heart of why polynomial rings over nice rings are themselves nice.

Call a polynomial with coefficients in a UFD R primitive if the greatest common divisor of its coefficients is a unit — that is, no irreducible of R divides all its coefficients. Gauss's lemma states that the product of two primitive polynomials is again primitive. From this it follows that a primitive polynomial which is irreducible over R remains irreducible over the field of fractions Frac(R), and conversely, so factoring over R and over Frac(R) agree for primitive polynomials.

The headline corollary is that if R is a UFD then so is R[x], and by induction R[x_1, ..., x_n]. This is precisely why Z[x] and k[x_1, ..., x_n] enjoy unique factorization even though they fail to be PIDs. The lemma also underlies practical irreducibility tests: to show a polynomial over Z is irreducible over Q, it suffices to show the primitive version is irreducible over Z, which is often far more tractable.