Sorry for my bad English.
Let $K\subset L$ be field extension such that transcendence degree tr.deg $L/K<\infty$.
Now, can we say $L/K$ is finitely generated field extension?
Please tell me proof or contradiction, thanks.
2026-04-30 00:09:27.1777507767
finite transcendence degree but not finitely generated field extension
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No, for example the algebraic closure $L$ of $\Bbb Q$ is algebraic over $\Bbb Q$, hence $L/\Bbb Q$ has transcendence degree $0$ but it cannot be finitely generated as it is an infinite algebraic extension.