Let $K$ be a field, and let $f,g\in K[Y]$ be coprime with $\deg(fg)\geq1$. How do I prove that $f-gX\in K(X)[Y]$ is irreducible?
I tried the "generic" approach of assuming the existence of a decomposition $f-gX=ab$ with non-units $a,b\in K(X)[Y]$, but this quickly exploded when trying to derive a contradiction.
Any advice is much appreciated.
Gauss' lemma applies: if $f-gX$ is reducible in $K(X)[Y]$, then it's reducible in $K[X][Y]$. Why does it apply? Because $K[X]$ is a principal ideal domain.