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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. $\square$

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Comment #1680 by Michele Serra on

Suggested slogan: A finitely generated algebraic extension is finite.

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