I'm stuck with a theorem I'm trying to prove, which says that for any $f\in \mathbb{Z}[x]$, and $g,h\in \mathbb{Q}[x]$ s.th. $f=gh$, there is some $\alpha \in \mathbb{Q}$ such that both $\alpha g$ and $\frac{1}{\alpha}h$ are in $\mathbb{Z}[x]$. It seems like just a few lines, and I tried to experiment with some of the functions being primitive, but I am not getting anywhere. What is the trick here?
reducing polynomials in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$
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abstract-algebra
polynomials
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0well, it is true as currently written. in that example, $gh=\frac{1}{5}x \frac{2}{3}x = \frac{2}{15}x^2\notin \mathbb{Z}[x]$. But thanks for the hint on the name!. – 2011-11-22