If $f,g$ in $Z[x]$, $h$ in $R[x]$ with $f=gh$, is $h$ nessecarily in $Z[x]$?

Question

Let $f$ and $g$ be monic polynomials in $Z[x]$. There exists a polynomial $h$ in $R[x]$ such that $f=gh$ for all real $x$. Is $h$ nessecarily in $Z[x]$?

Answer 1

The division algorithm can be performed in the rational numbers, so the coefficients of $h$ are rational. Write $$ h(x)=\frac{a}{b}(\underbrace{c_0+c_1x+\dots+c_nx^n}_{h_1(x)}) $$ where the greatest common divisor of the coefficients $c_0,c_1,\dots,c_n$ is $1$ and also $a$ and $b$ are coprime. It's not restrictive to assume $a>0$ and $b>0$.

Thus we have $$ bf(x)=ag(x)h_1(x) $$ and Gauss' lemma implies that $a=b$, since $f$ and $g$ are monic. So $f(x)=g(x)h_1(x)$ and therefore $h_1=h$.