Definition:Finitely Generated Field Extension

Definition
Let $E / F$ be a field extension.

Then we say $E$ is finitely generated over $F$ if $E = F \left({\alpha_1,\ldots,\alpha_n}\right)$ for some $\alpha_1,\ldots,\alpha_n \in E$.

Here $F(\alpha_1,\ldots,\alpha_n)$ is the field generated by $F \cup \{\alpha_1,\ldots,\alpha_n\}$