Separable polynomial

Definition

Let $K$ be a field and $f(x)\in K[x]$ be a nonzero polynomial. We say that $f$ is a separable polynomial if the following equivalent conditions are satisfied:

1. $f(x)$ and its formal derivative $f^{\prime }(x)$ are relatively prime in $K[x]$.
2. $f(x)$ splits completely into distinct linear factors over its splitting field.
3. For any field $L$ containing $K$, $f$ is a square-free polynomial over $L$, i.e., no square of a polynomial divides $f$.
4. The discriminant of $f$ is nonzero.