In section 4 of Samuel's Unique Factorization it's said Gauss's lemma is an easy consequence of Nagata's lemma. How to deduce Gauss's lemma from Nagata's lemma?
I'm asking because Nagata's lemma help to determine when irreducibles are primes, yet Gauss's lemma does not assume anything about irreducibility in $\mathbb Z[x]$ but rather obtains it from $\mathbb Q[x]$ and primitivity, a property that doesn't seem to fit in...
By Gauss' lemma he means the fact that if $A$ is factorial, then so is $A[T]$. Samuel explains the proof in his paper.... namely let $S=A-\{0\}\subset A[T]$. Then $S$ is generated by prime elements, and $S^{-1}A[T]=\mathrm{Frac}(A)[T]$ is a PID, hence factorial. Thus $A[T]$ is itself factorial.