Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$ The question as written in the title:-

Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$.

My attempt:- I know is $g=\gcd(f,f^{'})$ is not constant and hence separable. but how to go from here?

Almost there, if $f$ non-zero had a multiple root $a$ then $a$ would be a root of $f'$ too, so $(X-a)$ divides both $f$ and $f'$, so it divides $g = \gcd(f,f')$ that is not therefore a constant polynomial, and moreover, has degree $\le \deg f' < deg f$. So $g\in \mathbb{Q}[X]$ is a proper factor of $f$ and therefore, $f$ is reducible.
${\bf Added:}$ So if $f = c \prod (X-a_i)^{m_i}$  ( $ c \ne 0$ ) then 
$g = \prod (X-a_i)^{m_i-1}$ and $g \in \mathbb{Q}[X]$  ( the fact that $g \in \mathbb{Q}[X]$ is a consequence of Euclid's algorithm that takes place in $\mathbb{Q}[X]$ since both $f$ and $f'$ are there). 
