Let $k$ be a field and $\bar{k}/k$ be an algebraic closure of $k$. Let $k'$ be a finitely generated $k$-algebra, which is a subalgebra of $\bar{k}$. Is it true that $k'$ is a finite field extension of $k$?
Asked
Active
Viewed 34 times
1 Answers
2
Yes. $k'$ is finitely generated and integral over $k$, hence finite over $k$ (Stacks/02JJ). It is then a finite-dimensional $k$-vector space and an integral domain, hence a field (Stacks/00GS).
Martin Brandenburg
- 163,620
-
(+1) Perhaps more readable than the second link: https://math.stackexchange.com/a/4357501/386889 – Anne Bauval Dec 17 '22 at 23:27
-
1Yes. I assume this has been proven > 20 times on MSE. :-) Cheers! – Martin Brandenburg Dec 17 '22 at 23:29