A polynomial is separable if and only if $\gcd(f, f') = 1$

A polynomial is separable exactly when it is coprime to its derivative; their gcd records repeated roots.Suggest a slogan

Let $k$ be a field and $f \in k[x]$ a nonzero polynomial. Then $f$ is separable if and only if $\gcd(f, f') = 1$ in $k[x]$, that is, if and only if $f$ and $f'$ have no common root in an algebraic closure $\bar{k}$. Moreover, the repeated roots of $f$ in $\bar{k}$ are exactly the roots of $\gcd(f, f')$.