I am trying to prove the following statement.
Let $R$ be an integral domain. Prove that a non-constant irreducible polynomial in $R[x]$ is primitive.
This is what I have done so far:
Let $f\in R[x]$ be irreducible. Then there exists $g,h\in R[x]$ such that
$f=gh \implies f\sim g$ or $f\sim h$
Wlog, assume $f\sim g \implies h\in R[x]^X$
Let $g=a_nx^n+...+a_0$, where $n>0$
Thus, $C(f)=C(gh)=gcd(ha_n,...,ha_0)$ so $h$ is a common divisor for $f$.
It is at this stage that I don't know how to proceed further with the proof. Any help would be appreciated.