Gauss lemma for polynomials

Over a UFD, products of primitive polynomials are primitive and factorization over the fraction field descends after content is removed.

If $R$ is a unique factorization domain with fraction field $F$, then $R[X]$ is a unique factorization domain. A primitive polynomial irreducible in $R[X]$ remains irreducible, hence prime, in the principal ideal domain $F[X]$; Gauss lemma for polynomials brings divisibility back to $R[X]$.

The polynomial ring $\mathbb{Z}[X]$ is a unique factorization domain by Gauss lemma for polynomials, but its ideal $(2,X)$ is not principal. Any generator would divide both 2 and $X$, hence be a unit: a divisor of 2 has degree zero, and a constant dividing $X$ is $\pm 1$. Yet evaluation at zero followed by reduction modulo 2 annihilates the ideal and not 1. It embeds in the principal ideal domain $\mathbb{Q}[X]$, illustrating that the property need not pass to a subring.

Every rational function in $\mathbb{C}(X,Y)$ fixed by exchanging $X$ and $Y$ is a quotient of coprime symmetric polynomials. In a coprime presentation $p/q$, symmetry givesUnique factorization implies $p^{\ast }=\lambda p$ and $q^{\ast }=\lambda q$. The alternative $\lambda =-1$ would make both divisible by $X-Y$, contradicting coprimality, so $\lambda =1$.