# Definition:Finitely Generated Field Extension

## Definition

Let $E / F$ be a field extension.

Then $E$ is said to be finitely generated over $F$ if and only if, 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 $F \cup \left\{{\alpha_1, \ldots, \alpha_n}\right\}$.