Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in $K[t,x]$.
Does there exist $a \in K[t]$ such that $a f$ is irreducible in $K[t,x]$?