I have a polynomial $p$ with integer coefficients, which might have roots of multiplicity higher than one. I would like to obtain each root just once so I calculate $g = \gcd(p, p')$ and divide $q = p/g$. I know $g$ has integer coefficients, too, so my question is: does that division keep $q$ also in $\Bbb Z$?
I tried many random polynomials and it appears to be the case, but how can I be certain that this will always be the case? For example:
$$p = (x-7)\cdot(x-8)\cdot(x-8)\cdot(x-9)\cdot(x-9)\cdot(x-9)$$ $$p = x^6 - 50x^5 + 1040x^4 - 11518x^3 + 71631x^2 - 237168x + 326592$$
$$\gcd(p, p') = x^3 - 26x^2 + 225x - 648$$
$$p/q = x^3 - 24x^2 + 191x - 504$$