The Stacks project

A finitely generated algebraic extension is finite.

Lemma 9.8.6. Let $k$ be a field, and let $\alpha _1, \alpha _2, \ldots , \alpha _ n$ be elements of some extension field such that each $\alpha _ i$ is algebraic over $k$. Then the extension $k(\alpha _1, \ldots , \alpha _ n)/k$ is finite. That is, a finitely generated algebraic extension is finite.

Proof. Indeed, each extension $k(\alpha _{1}, \ldots , \alpha _{i+1})/k(\alpha _1, \ldots , \alpha _{i})$ is generated by one element and algebraic, hence finite. By multiplicativity of degree (Lemma 9.7.7) we obtain the result. $\square$


Comments (1)

Comment #1680 by Michele Serra on

Suggested slogan: A finitely generated algebraic extension is finite.

There are also:

  • 2 comment(s) on Section 9.8: Algebraic 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Your email address will not be published.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 09GH. Beware of the difference between the letter 'O' and the digit '0'.