Rational polynomial content ($c(g)=c(ng)/n\in {\mathbb{Q}}_{>0}$)

For a nonzero polynomial $g\in \mathbb{Q}[t]$, choose a positive integer $n$ clearing its denominators and set $c(g)=c(ng)/n$, where the polynomial content of an integer-coefficient polynomial is the positive greatest common divisor of its coefficients. This is independent of $n$: if $f/n=h/m$, then $mf=nh$, hence $mc(f)=nc(h)$.

The integer-coefficient product formula from Gauss lemma for polynomials givesMoreover $g/c(g)$ is an integer-coefficient primitive polynomial: if $g=f/n$, then $g/c(g)=f/c(f)$. For example,This normalization proves a useful divisibility descent. If $f\in \mathbb{Z}[t]$ is primitive, $g\in \mathbb{Z}[t]$ and $g=fq$ for $q\in \mathbb{Q}[t]$, then $c(q)=c(g)$ is an integer. Since $q/c(q)\in \mathbb{Z}[t]$, it follows that $q\in \mathbb{Z}[t]$.