Definition. separable extensions [1703]

Let $F$ be a field and $\overline{F}$ be an algebraic closure of $F$.

1. A polynomial in $F[x]$ is called separable if all its roots in $\overline{F}$ are simple. A polynomial which is not separable is called inseparable.
2. Let $K / F$ be a field extension and $\alpha \in K$ be an element algebraic over $F$. We say $\alpha$ is separable over $F$ if its minimal polynomial over $F$ is separable.
3. An algebraic extension $K / F$ is called separable if every element in $K$ is separable over $F$. An algebraic extension $K / F$ is called inseparable if it is not separable.