Monic irreducible integer-polynomial quotient is a domain ($\mathbb{Z}[X]/(p)$)

For monic $p\in \mathbb{Z}[X]$ irreducible in $\mathbb{Q}[X]$, monic division gives an injection $\mathbb{Z}[X]/(p)\hookrightarrow \mathbb{Q}[X]/(p)$: an integer polynomial in the rational ideal has zero integer remainder. The target is a field, so the source is an integral domain. For positive degree $d$, unique remainders make the source a free abelian group on $1,X,\dots ,X^{d-1}$. The nonzero integer $2$ is not a unit, since multiplying an integer remainder by two cannot give remainder one. Thus this integral domain is not a field.