Irreducibility of polynomial P(X) = X

Question

I have the following exercise:

Let $R$ be the integral domain of all polynomials $P(X)$ with real coefficients whose constant term is a rational. Is the poly $P(X) = X$ irreducible in $R$?

My question is: Is this similar to showing that $P(X+c)$ is irreducible if and only if $P(X)$ is irreducible. I do not understand the role on the constant term being rational.

Answer 1

Yes. First, observe $\mathbb{R}[X]^{\times} = \mathbb{R}^{\times}$ and $R^{\times} = \mathbb{Q}^{\times}$. Notice that $X$ is a non-zero non-unit in $R$. If $X = A \cdot B$ with $A,B \in R$, this equation also holds in $\mathbb{R}[X]$. Since $X$ is irreducible there, we get $A \in \mathbb{R}^{\times}$ or $B \in \mathbb{R}^{\times}$. The claim now follows from $\mathbb{R}^{\times} \cap R = \mathbb{Q}^{\times} = R^{\times}$.