Definition 9.6.6. Let $k$ be a field. If $F/k$ is an extension of fields and $S \subset F$, we write $k(S)$ for the smallest subfield of $F$ containing $k$ and $S$. We will say that $S$ *generates the field extension* $k(S)/k$. If $S = \{ \alpha \} $ is a singleton, then we write $k(\alpha )$ instead of $k(\{ \alpha \} )$. We say $F/k$ is a *finitely generated field extension* if there exists a finite subset $S \subset F$ with $F = k(S)$.

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