Lemma 9.7.5. A finite extension of fields is a finitely generated field extension. The converse is not true.

**Proof.**
Let $F/E$ be a finite extension of fields. Let $\alpha _1, \ldots , \alpha _ n$ be a basis of $F$ as a vector space over $E$. Then $F = E(\alpha _1, \ldots , \alpha _ n)$ hence $F/E$ is a finitely generated field extension. The converse is not true as follows from Example 9.7.4.
$\square$

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