If $f(X), g(X) \in \mathbb{Q} [X]$ and $f(X) = g(X) h(X)$ , is $h(X)\in\mathbb{Q} [X]$ ?
Probably this is really really basic, but just in case I am missing something...
I think that in general $h$ may not be a polynomial, but if it is then is has to be a rational one. Is that right? Any ideas how to prove it?
$f(X) = g(X) h(X)$ gives a system of linear equations for the coefficients of $h$. If this system has a solution, than it has to remain in $\mathbb Q$ because it can be found by Gaussian elimination.