Separable ⇔ Distinct Roots

Theorem.
Let $K$ be a field and $f\in K[x]$ be irreducible. The following are equivalent:

1. $f$ has distinct roots in an algebraic closure of $K$.
2. $\gcd(f,f')=1$ in $K[x]$, where $f'$ is the formal derivative.
3. Any (equivalently, every) root $\alpha$ of $f$ is a separable element over $K$.

In particular, a finite extension $L/K$ is separable iff every element of $L$ is separable over $K$.

Examples.

1. Over $\mathbb{Q}$, $f=x^3-2$ has derivative $3x^2$, and $\gcd(f,f')=1$, so $f$ is separable (its three roots are distinct).
2. Over $\mathbb{F}_p(t)$, $f=x^{p}-t$ has derivative $0$, hence is inseparable; it has a single root of multiplicity $p$ in an algebraic closure.
3. Over $\mathbb{F}_p$, $f=x^{p}-x$ satisfies $f'=-1\neq 0$, hence has no repeated roots and splits with $p$ distinct roots (namely all elements of $\mathbb{F}_p$).

Related. irreducible polynomials , inseparable extensions .