Definition:Finitely Generated Field Extension

Definition
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 $F \cup \left\{{\alpha_1, \ldots, \alpha_n}\right\}$.

Also see

 * Definition:Finite Field Extension