derivative of $f(x)$ where $F$ is any field. Then $f(x)$ has multiple roots if and only if $\gcd(f(x),f'(x))\ne 1$. If $F$ is a field of characteristic zero then you know more: If you denote $g(x)=\gcd(f(x),f'(x))$ and divide $f(x)$ by $g(x)$, i.e. you find the polynomial $h(x)$ such that $f(x)=g(x)\cdot h(x)$, then