Let $R = \mathbb{Z} + x \mathbb{Q}[x]$ be a ring. I want to compute the list of irreducible elements of $R.$
Claim: The set of all irreducible elements of $R$ is the set of prime numbers of $ \mathbb{N}$ and their negatives along with the irreducible polynomials in $\mathbb{Q}[x]$ with the constant term $1$ and -$1.$
If the constant term of the polynomial is an integer which is neither $1$ nor -$1,$ then choose $p(x) = a + xp^\prime(x).$ If $|a| \neq 1,$ then there exists a prime number m dividing $a$; say $a = mb.$ Then $p(x) = m(b + \frac{1}{m}xp^\prime(x)),$ where $p^\prime(x)$ is an element of $\mathbb{Q}[x].$ Since neither of these factors is a unit in $R$, so that p(x) is not irreducible. Hence, $|a| = 1$ is a necessary condition. By the way, I have shown that the units of $R$ are $1$ and $-1$. Can anyone help me to complete my proof? Thanks so much.