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Definition: A polynomial $f\in \mathbb{Z}[x]$ with $f(x)=a_0+a_1x+\dots+a_nx^n$ is a primitive polynomial the only common factors of $a_0,a_1,\dots,a_n$ are the units $\pm 1\in \mathbb{Z}$ and if $a_n>0$.

I stumbled across this sentence in my course notes, but I can't find a reasonable explanation for it:

Every polynomial $f\in\mathbb{Q}[x]$ can be uniquely written as a product $f(x)=c\cdot f_0(x)$ with $c\in\mathbb{Q}$ and $f$ a primitive polynomial in $\mathbb{Z}[x]$.

Thanks.

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  • $\begingroup$ Not quite: if $f$ is primitive, so is $-f$. So, for uniqueness you'd have to insist that $c$ is a positive rational. $\endgroup$ – Angina Seng Dec 24 '19 at 13:52
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We can multiply all the coefficients of $f$ by an integer so that they are all integers and so that the coefficient of the highest power of $x$ is positive.Then any common factor of all the coefficients can be factored out. The effect of this is that a rational times $f$ is a primitive polynomial.

If the ratio of two primitive polynomials were a rational, $a/b$ say, then let the polynomial $g$ be $a$ times one and also $b$ times the other. Then the g.c.d. of all the terms of $g$ would be $a$ and also would be $b$. Therefore $a=b$ and the representation is unique.

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  • $\begingroup$ Or $a=-b{}{}{}$? $\endgroup$ – Angina Seng Dec 24 '19 at 14:02
  • $\begingroup$ No - see the definition of primitive with $a_n$ positive. $\endgroup$ – S. Dolan Dec 24 '19 at 14:03

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