"Let $D$ be a UFD and let $F$ be its quotient field. Further let $f$ be a primitive polynomial of positive degree in $D\left[x\right]$. From this it follows that that $f$ is irreducible in $D\left[x\right]$ if and only if $f$ is irreducible in $F\left[x\right]$."
I've shown the forward direction, i.e. irreducible in UFD $\Rightarrow$ irreducible in quotient field, but am struggling to understand how the converse direction would go.
In particular, it strikes me that it might be productive to try and show that $f$ is prime in the UFD since this is equivalent to irreducibility, but I don't know how to show this. Also, $f$ should have no "denominators" in $F\left[x\right]$ since it's also in $D\left[x\right]$.
Any advice on how to proceed?