The Stacks project

Lemma 9.12.3. Let $K/E/F$ be a tower of algebraic field extensions.

  1. If $\alpha \in K$ is separable over $F$, then $\alpha $ is separable over $E$.

  2. if $K$ is separable over $F$, then $K$ is separable over $E$.

Proof. We will use Lemma 9.12.1 without further mention. Let $P$ be the minimal polynomial of $\alpha $ over $F$. Let $Q$ be the minimal polynomial of $\alpha $ over $E$. Then $Q$ divides $P$ in the polynomial ring $E[x]$, say $P = QR$. Then $P' = Q'R + QR'$. Thus if $Q' = 0$, then $Q$ divides $P$ and $P'$ hence $P' = 0$ by the lemma. This proves (1). Part (2) follows immediately from (1) and the definitions. $\square$

Comments (0)

There are also:

  • 7 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Your email address will not be published.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 09H2. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 9.12.3 as pdf