Separable element

Definition. Let $L/K$ be a field extension and let $\alpha\in L$ be algebraic over K . The element $\alpha$ is separable over $K$ if its minimal polynomial $m_\alpha(x)\in K[x]$ has distinct roots in a splitting field (equivalently, $m_\alpha$ has no repeated root).

A standard criterion is: $\alpha$ is separable iff $\gcd(m_\alpha, m_\alpha')=1$ in $K[x]$, where $m_\alpha'$ is the formal derivative.

See also. separable extension , inseparable extension , separable ⇔ distinct roots .

Examples.

1. Over any field of characteristic $0$ (e.g. $\mathbb{Q}$), every algebraic element is separable; in particular $\sqrt2$ is separable over $\mathbb{Q}$.
2. In $\mathbb{F}_p(t^{1/p})/\mathbb{F}_p(t)$, the element $t^{1/p}$ is algebraic but not separable: its minimal polynomial is $x^p-t$, which has repeated roots in characteristic $p$.
3. Every element of a finite field $\mathbb{F}_{p^n}$ is separable over $\mathbb{F}_p$.