Consider a factorial domain $R$. A polynomial $f \in R[T]$ is called a primitive polynomial ..
(a) if there is no irreducible element $p \in R$ that divides all the coefficients of $f$
(b) if 1 is a greatest common divisor of the coefficients of $f$
These two definitions are considered to be equivalent to one another.
I'm having some trouble proving this;
(b) $\implies$ (a) is pretty straightforward via contradiction: assume that there exists an irreducible element that divides all the coefficients of $f$; then by (a), this element must also divide 1 and thus it is a unit in $R$ (which contradicts (b) since a unit can not be irreducible)
(a) $\implies$ (b) seems harder to me; (a) implies that
either $p$ is a unit and thus (a) implies (b)
OR
p is reducible (and thus (b) cannot hold)
What am I doing wrong here? Any help is appreciated!