= Rational polynomial content
{title2=$c(g)=c(ng)/n\in\mathbb Q_{>0}$}
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> gives
$$
c(g_1g_2)=c(g_1)c(g_2),\qquad
c(rg)=|r|c(g)\quad(r\in\mathbb Q\setminus\{0\}).
$$
Moreover $g/c(g)$ is an integer-coefficient <primitive polynomial>: if $g=f/n$, then $g/c(g)=f/c(f)$. For example,
$$
g(t)=\frac23t^2-\frac49t+\frac29
\quad\Longrightarrow\quad
c(g)=\frac29,\qquad g/c(g)=3t^2-2t+1.
$$
This normalization proves a useful divisibility descent. If $f\in\mathbb Z[t]$ is <primitive polynomial>[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 $q\in\mathbb Z[t]$.
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