What are the irreducible elements of the polynomial ring $\mathbb Q[x]$?

Question

Let $R$ be the ring of $f(x) \in \mathbb{Q}[x]$ such that $f(0) \in \mathbb{Z}$. Find the irreducible elements of $R$.

I believe I need to prove three things. First, that elements of the form $\pm p$ for $p$ prime and functions $f(x)$ where $f(0) = \pm 1$ are irreducible. Second, I need to show that non-constant polynomials where $f(0) \neq \pm 1$ are not irreducible. Third, I need to show that constant polynomials that are not prime and not $\pm 1$ are reducible.

The last one is straightforward. If $f(x) = c$, then $f(x) = ab$ where $a,b$ are neither $c$ nor $1$ or $-1$. Since $\pm 1$ are the only units in $R$, neither $a$ nor $b$ are units, so $f(x)$ is reducible.

I can't figure out how to prove the first two parts or if I'm even on the right track.

Answer 1

First, if $f(0)=0$, then $f/2\in R$ hence $f=2\frac f2$ is reducible.

Similarly if $|f(0)|>1$, then $f/f(0)\in R$ and thus $f$ is reducible, unless it's constant, when it needs to be prime in order to be irreducible.

Now prove that if $f(0)=\pm1$ then $f$ is irreducible in $R$ iff it's irreducible in $\Bbb Q[x]$ (which is not necessarily straightforward to decide, by the way).

If $f=gh$ in $R$ then $g$ and $h$ are not constant (unless $\pm1$) since $f(0)=\pm1$, so it shows reducibility in $\Bbb Q[x]$.
For the converse, if $f=gh$ with nonconstant $g,h\in\Bbb Q[x]$ and $f\in R,\,f(0)=\pm1$, then we have $f(0)=g(0)h(0)=\pm1$ and thus $f=f(0)\cdot g/g(0)\cdot h/h(0)$ is a decomposition of $f$ in $R$.