= Monic irreducible integer-polynomial quotient is a domain
{title2=$\mathbb Z[X]/(p)$}
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 $1,X,\ldots,X^{d-1}$. 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.
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