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?

If $f$ is irreducible and has a multiple root $\alpha \in \mathbb{C}$ then $\alpha$ is also a zero of $f'$. Hence $f$ divides $f'$. But $deg(f')<deg(f)$ thus $f'=0$. This implies that $f$ is constant.
