The Stacks project

Lemma 9.12.11. Let $K/F$ be a finite extension of fields. Let $\overline{F}$ be an algebraic closure of $F$. Then we have

\[ |\mathop{\mathrm{Mor}}\nolimits _ F(K, \overline{F})| \leq [K : F] \]

with equality if and only if $K$ is separable over $F$.

Proof. This is a corollary of Lemma 9.12.10. Namely, since $K/F$ is finite we can find finitely many elements $\alpha _1, \ldots , \alpha _ n \in K$ generating $K$ over $F$ (for example we can choose the $\alpha _ i$ to be a basis of $K$ over $F$). If $K/F$ is separable, then each $\alpha _ i$ is separable over $F(\alpha _1, \ldots , \alpha _{i - 1})$ by Lemma 9.12.3 and we get equality by Lemma 9.12.10. On the other hand, if we have equality, then no matter how we choose $\alpha _1, \ldots , \alpha _ n$ we get that $\alpha _1$ is separable over $F$ by Lemma 9.12.10. Since we can start the sequence with an arbitrary element of $K$ it follows that $K$ is separable over $F$. $\square$

Comments (0)

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Your email address will not be published.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