= Solution
A <unique factorization domain> is an <integral domain> in which every nonzero nonunit is a finite product of <irreducible elements>, uniquely up to reordering and multiplication of factors by a <unit in a ring>. A <principal ideal domain> is an <integral domain> in which every <ideal> is generated by one element.
<Gauss lemma for polynomials> says that over a <unique factorization domain> $D$, the product of two <primitive polynomials> is primitive, and the <polynomial content> satisfies $c(fg)=c(f)c(g)$ up to a unit. In particular a primitive polynomial of positive degree is irreducible over $D$ exactly when it is irreducible over the <fraction field> of $D$. Consequently a <polynomial ring over a unique factorization domain> is itself a <unique factorization domain>.
The <Eisenstein criterion> says that if $f(X)=a_nX^n+\cdots+a_0\in D[X]$ has $n\ge1$ and there is a prime element $p$ with $p\nmid a_n$, $p\mid a_j$ for every $j<n$, and $p^2\nmid a_0$, then $f$ is irreducible over the <fraction field>. If $f$ is primitive, it is also irreducible in $D[X]$. Over $\mathbb Z$ one may take $p$ to be an ordinary prime number. The primitivity qualification prevents a nonunit constant content from being overlooked.
\b[The defining distinction is unique irreducible factorization versus generation of every ideal by one element.]
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