Gauss's Lemma (polynomial) - Proof of The Irreducibility Statement

Proof of The Irreducibility Statement

We prove the irreducibility statement directly in the setting of a UFD R. As mentioned above a non-constant polynomial is irreducible in R if and only if it is primitive and not a product of two non-constant polynomials in F. Being irreducible in F certainly excludes the latter possibility (since those non-constant polynomials would remain non-invertible in F), so the essential point left to prove is that if P is non-constant and irreducible in R then it is irreducible in F.

Note first that in F\{0} any class of associate elements (whose elements are related by multiplication by nonzero elements of the field F) meets the set of primitive elements in R: starting from an arbitrary element of the class, one can first (if necessary) multiply by a nonzero element of R to enter into the subset R (removing denominators), then divide by the greatest common divisor of all coefficients to obtain a primitive polynomial. Now assume that P is reducible in F, so P = ST with S,T non-constant polynomials in F. One can replace S and T by associate primitive elements S′, T′, and obtain P = αS′T′ for some nonzero α in F. But S′T′ is primitive in R by the primitivity statement, so α must lie in R (if written as an irreducible fraction, its denominator has to divide all coefficients of S′T′ because αS′T′ lies in R, but that means the denominator is invertible in R), and the decomposition P = αS′T′ contradicts the irreducibility of P in R.