\begin{definition}[Definition:Finitely Generated Field Extension]
Let $E / F$ be a field extension.
Then $E$ is said to be '''finitely generated over $F$''' {{iff}}, for some $\alpha_1, \ldots, \alpha_n \in E$:
:$E = F \left({\alpha_1, \ldots, \alpha_n}\right)$ 
where $F \left({\alpha_1, \ldots, \alpha_n}\right)$ is the field in $E$ generated by $F \cup \left\{{\alpha_1, \ldots, \alpha_n}\right\}$.
\end{definition}