## If the product of two rational polynomials is an integer polynomial, then the pairwise product of any two coefficients is an integer

Let $f(x), g(x) \in \mathbb{Q}[x]$ such that $f(x)g(x) \in \mathbb{Z}[x]$. Prove that the product of any coefficient of $f(x)$ with any coefficient of $g(x)$ is an integer.

Note that $f(x)g(x)$ is in $\mathbb{Z}[x]$ and factors in $\mathbb{Q}[x]$. By Gauss’ Lemma, there exist $r,s \in \mathbb{Q}$ such that $rf, sg \in \mathbb{Z}[x]$ and $(rf)(sg) = fg$. Since $\mathbb{Q}$ is an integral domain, in fact $rs = 1$. Let $f_i$ and $g_i$ denote the coefficients of $f$ and $g$, respectively; we have $rf_i \in \mathbb{Z}$ and $r^{-1}g_i \in \mathbb{Z}$, so that $f_ig_j \in \mathbb{Z}$ for all $i$ and $j$.

Note that the proof still works if we replace $\mathbb{Z}$ by an arbitrary unique factorization domain and $\mathbb{Q}$ by its field of fractions.

Look at the leading coefficient of $(rf)(sg) = fg$. (Perhaps a better justification would be ‘since $\mathbb{Q}$ is an integral domain’.)