Irreducible polynomial in characteristic zero is separable

If $f$ is irreducible over a field of characteristic zero, then $f^{\prime }\ne 0$ and $\mathrm{deg}{f}^{\prime }<\mathrm{deg}f$. Thus $f$ cannot divide $f^{\prime }$, so irreducibility gives $\gcd (f,f^{\prime })=1$. Consequently $f$ has no repeated root in any extension field.