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Show that any polynomial $p(x) \in \mathbb{Q}[x]$ can be written as $p(x) = tq(x)$ where $t \in \mathbb{Q}$ and $q(x) \in \mathbb{Z}[x]$ is primitive.

I started my proof by defining $p(x)$ as $(\frac{q}{r})_n x^n + \dots + (\frac{q}{r})_0$. Then I defined $t \in \mathbb{Q}$ as the product of greatest common factor of the coefficients of $p(x)$. I don't think this will work. How can I guarantee that after dividing $p(x)$ by $t$ I will get a primitive polynomial?

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You presumably brought the coefficients of $p(x)$ to some common denominator $r$, where $r$ is an integer. This can certainly be done.

So now the coefficients in the numerator are integers, say $b_n$ down to $b_0$. Let $d$ be the gcd of all of these, and let $c_i=b_i/d$. Then $p(x)$ is $\frac{d}{r}$ times the primitive polynomial $c_nx^n+\cdots +c_0$.

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