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".

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  • 9 comment(s) on Section 9.26: Transcendence

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