Separable element

Let $K/F$ be a field extension and let $\alpha\in K$ be an algebraic element over $F$. Let $m_{\alpha,F}(x)\in F[x]$ be the minimal polynomial of $\alpha$ over $F$.

Definition (separable element). The element $\alpha$ is separable over $F$ if the polynomial $m_{\alpha,F}(x)$ has no repeated roots in a splitting field (equivalently, in an algebraic closure of $F$). In other words, $m_{\alpha,F}(x)$ has $\deg(m_{\alpha,F})$ distinct roots.

A convenient algebraic criterion is:

where $m'_{\alpha,F}$ is the formal derivative. The link between separability and distinct roots is recorded in separable polynomials have distinct roots .

Examples.

1. Over $F=\mathbb{Q}$, $\alpha=\sqrt2$ is separable: its minimal polynomial $x^2-2$ has distinct roots $\pm\sqrt2$.
2. Over a field of characteristic $p>0$, separability can fail. In $K=\mathbb{F}_p(t^{1/p})$ over $F=\mathbb{F}_p(t)$, the element $\alpha=t^{1/p}$ has minimal polynomial $x^p-t$, whose derivative is $px^{p-1}=0$; the polynomial has a single root of multiplicity $p$, so $\alpha$ is not separable over $F$.
3. In any extension of a perfect field (e.g. $F=\mathbb{F}_p$ or $F=\mathbb{Q}$), every algebraic element is separable; in particular, every $\alpha\in \mathbb{F}_{p^n}$ is separable over $\mathbb{F}_p$.