How do you prove that, given a UFD polynomial ring, every irreducible polynomial is a primitive one. I am trying the following:
Let $R$ be a UFD, and $p\in R[X]$ be irreducible. We have the following:
$$p = c(p)\cdot p^*$$
where $c(p)\in R$, is a mcd of the coeficients of $p$, and $p^*$ is a primitive polynomial given by $p^* = p\cdot c(p)^{-1}$. Because $p$ is irreducible, then $c(p)$ is invertible or $p^*$ is invertible.
If $c(p)$ is invertible, we have by definition that $p$ is primitive.
What happens if $p^*$ is invertible?
I can't seem to find any argument to finish of the proof.