The following is written in Fulton's Algebraic curves book but I can't quite understand it.
If $R$ is a UFD with quotient field $K$, then (by Gauss) any irreducible element $F\in R[X]$ remains irreducible when considered in $K[X]$; it follows that if $F$ and $G$ are polynomials in $R[X]$ with no common factors in $R[X]$, they have no commmon factors in $K[X]$.
I understand Gauss' Lemma, but I can't see how it implies the sentence in boldface.