Definition 9.12.2. Let $F$ be a field. Let $K/F$ be an extension of fields.

1. We say an irreducible polynomial $P$ over $F$ is separable if it is relatively prime to its derivative.

2. Given $\alpha \in K$ algebraic over $F$ we say $\alpha$ is separable over $F$ if its minimal polynomial is separable over $F$.

3. If $K$ is an algebraic extension of $F$, we say $K$ is separable1 over $F$ if every element of $K$ is separable over $F$.

[1] For nonalgebraic extensions this definition does not make sense and is not the correct one. We refer the reader to Algebra, Sections 10.42 and 10.44.

Comment #4936 by Laurent Moret-Bailly on

The comment that the definition is restricted to algebraic extensions is welcome, but why not add a reference to sections 030I and 05DT?

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