Ytukyg

I need to prove that a polynomial $f left( x right) in mathbb{Q} left[ X right]$ is divisible by a square of a polynomial iff $f$ and $f'$ have a greatest common divisor of positive degree.

I have no idea where to start, except that I have to use the properties of Polynomial Rings as Euclidean Domain.

Let $q(x)$ be an irreducible common divisor of $f$ and $f'$. Division of $f$ by $q^2$ provides
$$
f=kq^2+r, quad text{with},,,deg r<deg q^2
$$
and as $q$ divides $f$, then $q$ divides $r$, and thus
$$
f=kq^2+r_1q, quad text{with},,,deg r_1<deg q.
$$
Differentiating we obtain
$$
f'=k'q^2+(2kq'+r_1')q+r_1q'
$$
But since $q$ divides $f'$, then $q$ divides $r_1q'$, and since $q$ is irreducible, then $q$ divides $q'$or $r_1$. But, $,deg q>deg q'$, and hence $q$ does not divide $q'$, as $q'ne 0$. Hence $q$ divides $r_1$, which implies that $r_1equiv 0$, since $deg r_1<deg q$.

Hint: One direction is easy. For the other direction, note that $f$ and $f'$ have a greatest common divisor of positive degree iff $f$ and $f'$ have an irreducible common divisor.

Make sure to pinpoint where working over $mathbb{Q}$ enters in your argument.