The Stacks project

Lemma 9.26.11. Let $K/k$ be a finitely generated field extension. The algebraic closure of $k$ in $K$ is finite over $k$.

Proof. Let $x_1, \ldots , x_ r \in K$ be a transcendence basis for $K$ over $k$. Then $n = [K : k(x_1, \ldots , x_ r)] < \infty $. Suppose that $k \subset k' \subset K$ with $k'/k$ finite. In this case $[k'(x_1, \ldots , x_ r) : k(x_1, \ldots , x_ r)] = [k' : k] < \infty $, see Lemma 9.26.10. Hence

\[ [k' : k] = [k'(x_1, \ldots , x_ r) : k(x_1, \ldots , x_ r)] \leq [K : k(x_1, \ldots , x_ r)] = n. \]

In other words, the degrees of finite subextensions are bounded and the lemma follows. $\square$


Comments (4)

Comment #4976 by Laurent Moret-Bailly on

I believe the equality needs some justification.

Comment #6317 by Peng DU on

I think the last "<" can be "\leq".

There are also:

  • 9 comment(s) on Section 9.26: Transcendence

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Your email address will not be published.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 037J. Beware of the difference between the letter 'O' and the digit '0'.