The Stacks project

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.

Comments (2)

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?

There are also:

  • 6 comment(s) on Section 9.12: Separable extensions

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09H1. Beware of the difference between the letter 'O' and the digit '0'.