If $D$ is a $UFD$ with fraction field $F$ why are nonzero nonconstant elements in $D[x]$ never primitive in $F[x]$ even if it is a primitive element of $D[x]$?
This is a question from my abstract algebra exam. This question confuses me. From my understanding, all nonzero nonconstant elements of $F[x]$ is primitive. That is because all nonzero elements in $F$ are considered unit, hence the gcd of the coefficients of the polynomials in $F[x]$ are any units of $F$.
Is my understanding wrong? I would truly appreciate it too if you could explain to me the answer to the question. Thanks!